ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 7 - 2016
Inference for Zero Inflated Truncated Power Series Family of Distributions
MK Patil*
Padmabhushan Vasantraodada Patil Mahavidyalaya, India
Received: August 14, 2016 | Published: December 06, 2016
*Corresponding author: MK Patil, Padmabhushan Vasantraodada Patil Mahavidyalaya, Kavathe Mahankal, Dist. Sangli, India, Email:
Citation: Patil MK (2016) Inference for Zero Inflated Truncated Power Series Family of Distributions. Biom Biostat Int J 4(7): 00115. DOI: 10.15406/bbij.2016.04.00115

Abstract

Zero-inflated data indicates that the data set contains an excessive number of zeros. The word zero-inflation is used to emphasize that the probability mass at the point zero exceeds than the one allowed under a standard parametric family of discrete distributions. Gupta et al. [1], Murat & Szynal [2], Patil & Shirke [3] have contributed to estimation and testing of the parameters involved in Zero Inflated Power Series Distributions. If the data set under study does not contain observations after some known point in the support, we have to modify Zero Inflated Power Series Distribution (ZIPSD) accordingly in order to get better inferential properties. Zero Inflated Truncated Power Series Distribution (ZITPSD) is one of the better options. In the present work we address problem of estimation for ZITPSD with more emphasis on statistical tests. We provide three asymptotic tests for testing the parameter of ZITPSD, using an unconditional (standard) likelihood approach, a conditional likelihood approach and the sample mean, respectively. The performance of first two tests has been studied for Zero Inflated Truncated Poisson Distribution (ZITPD). Asymptotic Confidence Intervals for the parameter are also provided. The model has been applied to a real life data.

Keywords: Zero Inflation; Zero Inflated Power Series Distribution; Zero Inflated Truncated Power Series Distribution; Zero Inflated Truncated Poisson Distribution

Introduction

In certain applications involving discrete data, we come across data having frequency of an observation ‘zero’ significantly higher than the one predicted by the assumed model. The problem of high proportion of zeros has been an interest in data analysis and modeling. There are many situations in the medical field, engineering applications, manufacturing, economics, public health, road safety epidemiology and in other areas leading to similar situations. In highly automated production process, occurrence of defects is assumed to be Poisson. However, we get no defectives in many samples. This leads to excess number of zeros. Models having more number of zeros significantly are known as zero-inflated models.

In the literature, numbers of researchers have worked on family of zero-inflated power series distributions. Gupta et al. [1] have studied the structural properties and point estimation of parameters of Zero-Inflated Modified Power Series distributions and in particular for zero-inflated Poisson distribution. Murat & Szynal [2] have studied the class of inflated modified power series distributions where inflation occurs at any of the support points. Moments, factorial moments, central moments, the maximum likelihood estimators and variance-covariance matrix of the estimators are obtained. Murat & Szynal [2] extended the results of Gupta et al. [1] to the discrete distributions inflated at any point s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@36EF@ .

Zero Inflated Truncated Power Series Distribution contains two parameters. The first parameter indicates inflation ( π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ ) of zero and the other parameter ( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ ) is that of power series distribution. Literature survey reveals that many researchers devoted to the inflation parameter of the model. In the present study, we focus on the referential aspect of the basic parameter of the model. In this article, we provide maximum likelihood parameters, Fisher information and asymptotic tests for testing the parameter of the Zero Inflated Truncated Power Series Distribution. Additionally, asymptotic confidence interval for the parameter is provided.

In section 2.1 we report estimation of both the parameters of ZITPSD and corresponding asymptotic variances using full likelihood approach, conditional likelihood approach and method of moments. In section 2.2, we provide three asymptotic tests for testing the parameter of ZITPSD. Section 2.3 is devoted to asymptotic confidence intervals for the parameters of ZITPSD. In section 3.1 we report estimation of parameters involved in Zero Inflated Truncated Poisson Distribution (ZITPD) and inference related to the model. Section 3.2 is devoted to three asymptotic tests for testing the parameter of ZITPD and in section 3.3 we provide asymptotic confidence intervals for the parameters of ZITPD. Simulation study is carried out in section 4, to study the performance of the tests. Illustrative example is provided in section 5.

Zero-Inflated Truncated Power Series Distribution(ZITPSD)

Before we define truncated ZIPSD, we first consider the Truncated Power Series Distribution (TPSD) truncated at the support point 't' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacEcaca WG0bGaai4jaaaa@38C8@  onwards, where 't' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacEcaca WG0bGaai4jaaaa@38C8@  is known. Then the probability mass function of TPSD is given by

P(X=x)= b x θ x f(θ)(1P(X>t) ,forx=0,1,2,...,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuai aacIcacaWGybGaeyypa0JaamiEaiaacMcacqGH9aqpdaWcaaqaaiaa dkgadaWgaaqcfasaaiaadIhaaKqbagqaaiabeI7aXnaaCaaabeqcfa saaiaadIhaaaaajuaGbaGaamOzaiaacIcacqaH4oqCcaGGPaGaaiik aiaaigdacqGHsislcaWGqbGaaiikaiaadIfacqGH+aGpcaWG0bGaai ykaaaacaGGSaGaaGzbVlaadAgacaWGVbGaamOCaiaaysW7caWG4bGa eyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikdacaGGSaGaaiOlai aac6cacaGGUaGaaiilaiaadshaaaa@5EF8@

  = b x θ x ( y=0 t b y θ y ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaamOyamaaBaaajuaibaGaamiEaaqcfayabaGaeqiUde3a aWbaaeqajuaibaGaamiEaaaaaKqbagaadaqadaqaamaaqahabaGaam OyamaaBaaajuaibaGaamyEaaqcfayabaGaeqiUde3aaWbaaeqajuai baGaamyEaaaaaeaacaWG5bGaeyypa0JaaGPaVlaaicdaaeaacaWG0b aajuaGcqGHris5aaGaayjkaiaawMcaaaaacaGGSaGaaGzbVlaaywW7 aaa@50F6@ = b x θ x G(θ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaamOyamaaBaaajuaibaGaamiEaaqcfayabaGaeqiUde3a aWbaaeqajuaibaGaamiEaaaaaKqbagaacaWGhbGaaiikaiabeI7aXj aacMcaaaGaaiilaaaa@426C@   where G(θ)= y=0 t b y θ y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeaca GGOaGaeqiUdeNaaiykaiabg2da9maaqahabaGaamOyamaaBaaajuai baGaamyEaaqcfayabaGaeqiUde3aaWbaaeqajuaibaGaamyEaaaaae aacaWG5bGaeyypa0JaaGPaVlaaicdaaeaacaWG0baajuaGcqGHris5 aaaa@491A@          

It is clear that the truncated distribution is also Power series distribution. Based on the same, we define ZITPSD as follows:

Let the probability mass function of a random variable X is given by

P(X=x)={ 1π+π b 0 G(θ)       for  x=0 π b x θ x G(θ)       for   x=1,2,3,...,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca GGOaGaamiwaiabg2da9iaadIhacaGGPaGaeyypa0Zaaiqaaqaabeqa aiaaigdacqGHsislcqaHapaCcqGHRaWkcqaHapaCdaWcaaqaaiaadk gadaWgaaqcfasaaiaaicdaaKqbagqaaaqaaiaadEeacaGGOaGaeqiU deNaaiykaaaaqaaaaaaaaaWdbiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOa8aacaWGMbGaam4BaiaadkhapeGaaiiOaiaacckapaGa amiEaiabg2da9iaaicdaaeaadaWcaaqaaiabec8aWjaadkgadaWgaa qcfasaaiaadIhaaKqbagqaaiabeI7aXnaaCaaabeqcfasaaiaadIha aaaajuaGbaGaam4raiaacIcacqaH4oqCcaGGPaaaa8qacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckapaGaamOzaiaad+gacaWGYbWd biaacckacaGGGcGaaiiOa8aacaWG4bGaeyypa0JaaGymaiaacYcaca aIYaGaaiilaiaaiodacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaa dshaaaGaay5Eaaaaaa@7C74@ …(2.1)

where G(θ)= y=0 t b y θ y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEeaca GGOaGaeqiUdeNaaiykaiabg2da9maaqahabaGaamOyamaaBaaajuai baGaamyEaaqcfayabaGaeqiUde3aaWbaaeqajuaibaGaamyEaaaaae aacaWG5bGaeyypa0JaaGPaVlaaicdaaeaacaWG0baajuaGcqGHris5 aaaa@491A@

Estimation of π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@

Estimation of the parameters using full likelihood function: Suppose a random sample X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada WgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaaMc8UaaGjbVlaadIfa daWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaaMe8UaaiOlaiaac6 cacaGGUaGaaiilaiaaysW7caWGybWaaSbaaKqbGeaacaWGUbaajuaG beaaaaa@4869@  of size n from ZITPSD is available. Then the likelihood function is given b

L(θ,π; x _ )= i=1 n ( 1π+ π b 0 G(θ) ) 1 a i ( πb x i θ x i G(θ) ) a i θ,π>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeaca GGOaGaeqiUdeNaaiilaiabec8aWjaaysW7caGG7aGaaGPaVpaamaaa baGaamiEaaaacaaMe8Uaaiykaiabg2da9maarahabaWaaeWaaeaaca aIXaGaeyOeI0IaeqiWdaNaey4kaSYaaSaaaeaacqaHapaCcaaMc8Ua amOyamaaBaaajuaibaGaaGimaaqcfayabaaabaGaam4raiaacIcacq aH4oqCcaGGPaaaaaGaayjkaiaawMcaaaqcfasaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqcfaOaey4dIunadaahaaqabKqbGeaacaaIXa GaeyOeI0IaamyyaKqbaoaaBaaajuaibaGaamyAaaqabaaaaKqbaoaa bmaabaWaaSaaaeaacqaHapaCcaaMc8UaamOyaiaadIhadaWgaaqcfa saaiaadMgaaKqbagqaaiaaykW7cqaH4oqCdaahaaqabKqbGeaacaWG 4bqcfa4aaSbaaKqbGeaacaWGPbaabeaaaaaajuaGbaGaam4raiaacI cacqaH4oqCcaGGPaaaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaa dggajuaGdaWgaaqcfasaaiaadMgaaeqaaaaajuaGcaaMf8UaaGzbVl abeI7aXjaaykW7caGGSaGaaGPaVlaaysW7cqaHapaCcqGH+aGpcaaI Waaaaa@83F1@   

where a i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada WgaaqcfasaaiaadMgaaKqbagqaaiabg2da9iaaicdaaaa@3AEA@  if x i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadMgaaKqbagqaaiaaysW7cqGH9aqpcaaIWaaaaa@3C8E@ and a i =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggada WgaaqcfasaaiaadMgaaKqbagqaaiabg2da9iaaigdaaaa@3AEB@  if x i = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhada WgaaqcfasaaiaadMgaaKqbagqaaiaaysW7cqGH9aqpaaa@3BD4@ 1,2,3,….t. …(2.2)

then, logL(θ,π; x _ )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGacYgaca GGVbGaai4zaiaadYeacaGGOaGaeqiUdeNaaiilaiabec8aWjaaysW7 caGG7aGaaGPaVpaamaaabaGaamiEaaaacaaMe8Uaaiykaiabg2da9a aa@470D@   = n 0 log( 1π+ π b 0 G(θ) )+ i=1 n a i logπ+ i=1 n a i logb x i + i=1 n a i x i log(θ) i=1 n a i logG(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9i aad6gadaWgaaqcfasaaiaaicdaaKqbagqaaiGacYgacaGGVbGaai4z amaabmaabaGaaGymaiabgkHiTiabec8aWjabgUcaRmaalaaabaGaeq iWdaNaaGPaVlaadkgadaWgaaqcfasaaiaaicdaaKqbagqaaaqaaiaa dEeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaacqGHRaWkda aeWbqaaiaadggadaWgaaqaaiaadMgaaeqaaiGacYgacaGGVbGaai4z aiabec8aWjabgUcaRaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaam OBaaqcfaOaeyyeIuoadaaeWbqaaiaadggadaWgaaqcfasaaiaadMga aKqbagqaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfa OaeyyeIuoaciGGSbGaai4BaiaacEgacaWGIbGaamiEamaaBaaajuai baGaamyAaaqcfayabaGaey4kaSYaaabCaeaacaWGHbWaaSbaaKqbGe aacaWGPbaajuaGbeaacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaa aKqbGeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLd GaciiBaiaac+gacaGGNbGaaiikaiabeI7aXjaacMcacqGHsisldaae WbqaaiaadggadaWgaaqcfasaaiaadMgaaKqbagqaaiGacYgacaGGVb Gaai4zaiaadEeacaGGOaGaeqiUdeNaaiykaaqcfasaaiaadMgacqGH 9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoacaaMf8UaaGzbVdaa@918F@ …(2.3)

Maximum likelihood estimators of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  and π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdaNaaG jbVlaaykW7aaa@3ACC@ are obtained by solving the following two equations

π ^ = (n n 0 )G( θ ^ ) n(G( θ ^ ) b 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0ZaaSaaaeaacaGGOaGaamOBaiabgkHiTiaad6gadaWg aaqcfasaaiaaicdaaKqbagqaaiaacMcacaWGhbGaaiikaiqbeI7aXz aajaGaaiykaaqaaiaad6gacaGGOaGaam4raiaacIcacuaH4oqCgaqc aiaacMcacqGHsislcaWGIbWaaSbaaKqbGeaacaaIWaaajuaGbeaaca GGPaaaaaaa@4CAC@                                                       …(2.4)

i=1 n a i x i θ ^ = n 0 π ^ b 0 G ( θ ^ ) G ( θ ^ ) 2 ( 1 π ^ + π ^ b 0 G( θ ^ ) ) + i=1 n a i G ( θ ^ ) G( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaabCaeaacaWGHbWaaSbaaKqbGeaacaWGPbaajuaGbeaacaWG4bWa aSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaaKqbakabggHiLdaabaGafqiUdeNbaKaaaaGaeyyp a0ZaaSaaaeaacaWGUbWaaSbaaKqbGeaacaaIWaaajuaGbeaacuaHap aCgaqcaiaaysW7caWGIbWaaSbaaKqbGeaacaaIWaaajuaGbeaaceWG hbGbauaacaGGOaGafqiUdeNbaKaacaGGPaaabaGaam4raiaacIcacu aH4oqCgaqcaiaacMcadaahaaqabKqbGeaacaaIYaaaaKqbaoaabmaa baGaaGymaiabgkHiTiqbec8aWzaajaGaey4kaSIafqiWdaNbaKaada WcaaqaaiaadkgadaWgaaqcfasaaiaaicdaaKqbagqaaaqaaiaadEea caGGOaGafqiUdeNbaKaacaGGPaaaaaGaayjkaiaawMcaaaaacqGHRa WkdaWcaaqaamaaqahabaGaamyyamaaBaaajuaibaGaamyAaaqcfaya baGabm4rayaafaGaaiikaiqbeI7aXzaajaGaaiykaaqcfasaaiaadM gacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoaaeaacaWGhbGa aiikaiqbeI7aXzaajaGaaiykaaaaaaa@7777@ , …(2.5)

Substituting π ^ = (n n 0 )G( θ ^ ) n(G( θ ^ ) b 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0ZaaSaaaeaacaGGOaGaamOBaiabgkHiTiaad6gadaWg aaqcfasaaiaaicdaaKqbagqaaiaacMcacaWGhbGaaiikaiqbeI7aXz aajaGaaiykaaqaaiaad6gacaGGOaGaam4raiaacIcacuaH4oqCgaqc aiaacMcacqGHsislcaWGIbWaaSbaaKqbGeaacaaIWaaajuaGbeaaca GGPaaaaaaa@4CAC@  in eq. (2.5) we get

x ¯ = θ ^ G ( θ ^ ) (G( θ ^ ) b 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qeaiabg2da9maalaaabaGafqiUdeNbaKaacaaMe8Uabm4rayaafaGa aiikaiqbeI7aXzaajaGaaiykaaqaaiaaysW7caGGOaGaam4raiaacI cacuaH4oqCgaqcaiaacMcacqGHsislcaWGIbWaaSbaaKqbGeaacaaI WaaajuaGbeaacaGGPaaaaaaa@4A2A@ ,                     …(2.6)

which is non-linear equation in θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ , Using Newton-Raphson method first we find θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@ , substituting this value of θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@  in Eq. (2.4) we find π ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaaaaa@3846@ . The Fisher information matrix of δ _ =(π,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaamaaaba GaeqiTdqgaaiabg2da9iaacIcacqaHapaCcaGGSaGaeqiUdeNabiyk ayaafaaaaa@3EBC@  is given by

I( δ _ )=( E( 2 logL π 2 ) E( 2 logL πθ ) E( 2 logL πθ ) E( 2 logL θ 2 ) )=( I 11 I 12 I 21 I 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeaca GGOaWaaWaaaeaacqaH0oazaaGaaiykaiabg2da9maabmaabaqbaeqa biGaaaqaaiabgkHiTiaadweadaqadaqaamaalaaabaGaeyOaIy7aaW baaeqajuaibaGaaGOmaaaajuaGciGGSbGaai4BaiaacEgacaWGmbaa baGaeyOaIyRaeqiWda3aaWbaaeqajuaibaGaaGOmaaaaaaaajuaGca GLOaGaayzkaaaabaGaeyOeI0IaamyramaabmaabaWaaSaaaeaacqGH ciITdaahaaqabKqbGeaacaaIYaaaaKqbakGacYgacaGGVbGaai4zai aadYeaaeaacqGHciITcqaHapaCcaaMc8UaeyOaIyRaeqiUdehaaaGa ayjkaiaawMcaaaqaaiabgkHiTiaadweadaqadaqaamaalaaabaGaey OaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGciGGSbGaai4BaiaacEga caWGmbaabaGaeyOaIyRaeqiWdaNaaGPaVlabgkGi2kabeI7aXbaaai aawIcacaGLPaaaaeaacqGHsislcaWGfbWaaeWaaeaadaWcaaqaaiab gkGi2oaaCaaabeqcfasaaiaaikdaaaqcfaOaciiBaiaac+gacaGGNb GaamitaaqaaiabgkGi2kabeI7aXnaaCaaabeqcfasaaiaaikdaaaaa aaqcfaOaayjkaiaawMcaaaaaaiaawIcacaGLPaaacqGH9aqpdaqada qaauaabeqaciaaaeaacaWGjbWaaSbaaKqbGeaacaaIXaGaaGymaaqc fayabaaabaGaamysamaaBaaajuaibaGaaGymaiaaikdaaKqbagqaaa qaaiaadMeadaWgaaqcfasaaiaaikdacaaIXaaajuaGbeaaaeaacaWG jbWaaSbaaKqbGeaacaaIYaGaaGOmaaqcfayabaaaaaGaayjkaiaawM caaaaa@8CE6@

Where

  I 11 =( n( G(θ) b 0 ) π( G(θ)πG(θ)+π b 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaaigdacaaIXaaajuaGbeaacqGH9aqpdaqadaqaamaa laaabaGaamOBamaabmaabaGaam4raiaacIcacqaH4oqCcaGGPaGaey OeI0IaamOyamaaBaaajuaibaGaaGimaaqcfayabaaacaGLOaGaayzk aaaabaGaeqiWda3aaeWaaeaacaWGhbGaaiikaiabeI7aXjaacMcacq GHsislcqaHapaCcaaMe8Uaam4raiaacIcacqaH4oqCcaGGPaGaey4k aSIaeqiWdaNaaGjbVlaadkgadaWgaaqcfasaaiaaicdaaKqbagqaaa GaayjkaiaawMcaaaaaaiaawIcacaGLPaaaaaa@5BD8@ ,                                           …(2.7)

I 12 =( n b 0 G (θ) G(θ)( G(θ)πG(θ)+π b 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaaigdacaaIYaaajuaGbeaacaaMc8Uaeyypa0ZaaeWa aeaadaWcaaqaaiaad6gacaaMe8UaamOyamaaBaaajuaibaGaaGimaa qcfayabaGabm4rayaafaGaaiikaiabeI7aXjaacMcaaeaacaWGhbGa aiikaiabeI7aXjaacMcadaqadaqaaiaadEeacaGGOaGaeqiUdeNaai ykaiabgkHiTiabec8aWjaayIW7caWGhbGaaiikaiabeI7aXjaacMca cqGHRaWkcqaHapaCcaaMi8UaamOyamaaBaaajuaibaGaaGimaaqcfa yabaaacaGLOaGaayzkaaaaaaGaayjkaiaawMcaaaaa@5EAD@                                          …(2.8)

and

I 22 =( nπ b 0 ( ( G (θ) 2 G (θ)2 G (θ) 2 G(θ) G (θ) 4 )+ π b 0 G (θ) 2 G (θ) 4 ( 1π+π b 0 G(θ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaaikdacaaIYaaajuaGbeaacqGH9aqpdaqabaqaaiaa d6gacaaMc8UaeqiWdaNaaGjbVlaadkgadaWgaaqcfasaaiaaicdaaK qbagqaamaabmaabaWaaeWaaeaadaWcaaqaaiaadEeacaGGOaGaeqiU deNaaiykamaaCaaabeqcfasaaiaaikdaaaqcfaOabm4rayaagaGaai ikaiabeI7aXjaacMcacqGHsislcaaIYaGabm4rayaafaGaaiikaiab eI7aXjaacMcadaahaaqabKqbGeaacaaIYaaaaKqbakaadEeacaGGOa GaeqiUdeNaaiykaaqaaiaadEeacaGGOaGaeqiUdeNaaiykamaaCaaa beqcfasaaiaaisdaaaaaaaqcfaOaayjkaiaawMcaaiabgUcaRmaala aabaGaeqiWdaNaaGjbVlaadkgadaWgaaqcfasaaiaaicdaaKqbagqa aiqadEeagaqbaiaacIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaG OmaaaaaKqbagaacaWGhbGaaiikaiabeI7aXjaacMcadaahaaqabKqb GeaacaaI0aaaaKqbaoaabmaabaGaaGymaiabgkHiTiabec8aWjabgU caRiabec8aWnaalaaabaGaamOyamaaBaaajuaibaGaaGimaaqcfaya baaabaGaam4raiaacIcacqaH4oqCcaGGPaaaaaGaayjkaiaawMcaaa aaaiaawIcacaGLPaaaaiaawIcaaaaa@8051@                                  

+nπ( G (θ) θG(θ) )+ nπ( 1 b 0 G(θ) )( G(θ) G (θ) G (θ) 2 ) G (θ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaKqbagaacqGHRaWkca WGUbGaaGPaVlabec8aWnaabmaabaWaaSaaaeaacaaMc8Uabm4rayaa faGaaiikaiabeI7aXjaacMcaaeaacqaH4oqCcaaMi8Uaam4raiaacI cacqaH4oqCcaGGPaaaaaGaayjkaiaawMcaaiabgUcaRmaalaaabaGa amOBaiaaykW7cqaHapaCdaqadaqaaiaaigdacqGHsisldaWcaaqaai aadkgadaWgaaqcfasaaiaaicdaaKqbagqaaaqaaiaadEeacaGGOaGa eqiUdeNaaiykaaaaaiaawIcacaGLPaaadaqadaqaaiaadEeacaGGOa GaeqiUdeNaaiykaiqadEeagaGbaiaacIcacqaH4oqCcaGGPaGaeyOe I0Iabm4rayaafaGaaiikaiabeI7aXjaacMcadaahaaqabKqbGeaaca aIYaaaaaqcfaOaayjkaiaawMcaaaqaaiaadEeacaGGOaGaeqiUdeNa aiykamaaCaaabeqcfasaaiaaikdaaaaaaaaa@6D21@  …(2.9)

 Assuming that conditions required for asymptotic normality for maximum likelihood estimators are satisfied, we have following theorem:

Theorem 2.1: Let X 1 , X 2 ,... X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada WgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWGybWaaSbaaKqbGeaa caaIYaaajuaGbeaacaGGSaGaaiOlaiaac6cacaGGUaGaamiwamaaBa aajuaibaGaamOBaaqcfayabaaaaa@4187@  be a random sample from ZITPSD with parameters π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ . Then the maximum likelihood estimator obtained by solving eq. (2.4) and eq. (2.6), have asymptotic bivariate normal distribution with mean vector (π,θ)' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcacq aHapaCcaGGSaGaeqiUdeNaaiykaiaacEcaaaa@3CA0@  and dispersion matrix I 1 ( δ _ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada ahaaqabKqbGeaacqGHsislcaaIXaaaaKqbakaacIcadaadaaqaaiab es7aKbaacaGGPaaaaa@3CDB@  for n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gaaa a@376C@  sufficiently large.

That is as n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gacq GHsgIRcqGHEisPaaa@3ACA@ , ( n ( π ^ π), n ( θ ^ θ) ) N 2 (0, I 1 ( δ _ )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaOaaaeaacaWGUbaabeaacaGGOaGafqiWdaNbaKaacqGHsislcqaH apaCcaGGPaGaaiilamaakaaabaGaamOBaaqabaGaaiikaiqbeI7aXz aajaGaeyOeI0IaeqiUdeNaaiykaaGaayjkaiaawMcaaiabgkziUkaa d6eadaWgaaqcfasaaiaaikdaaKqbagqaaiaacIcacaaIWaGaaiilai aadMeadaahaaqabKqbGeaacqGHsislcaaIXaaaaKqbakaacIcadaad aaqaaiabes7aKbaacaGGPaGaaiykaaaa@53C8@ .

In the following we present conditional likelihood approach and obtain MLEs for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ .

Conditional likelihood function approach: We observe that the conditional density of X i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada WgaaqcfasaaiaadMgaaKqbagqaaaaa@3921@  given A i = a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeada WgaaqcfasaaiaadMgaaKqbagqaaiabg2da9iaadggadaWgaaqcfasa aiaadMgaaKqbagqaaaaa@3CC1@  is independent of inflation parameter π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ , since

P( X i = x i | A i = a i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca GGOaGaamiwamaaBaaajuaibaGaamyAaaqcfayabaGaeyypa0JaamiE amaaBaaajuaibaGaamyAaaqcfayabaWaaqqaaeaacaWGbbWaaSbaaK qbGeaacaWGPbaajuaGbeaacqGH9aqpcaWGHbWaaSbaaKqbGeaacaWG PbaajuaGbeaaaiaawEa7aiaacMcaaaa@46F9@ = ( b x i (θ) x i G(θ) b 0 ) a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aabmaabaWaaSaaaeaacaaMc8UaamOyamaaBaaajuaibaGaamiEaKqb aoaaBaaajuaibaGaamyAaaqabaaajuaGbeaacaGGOaGaeqiUdeNaai ykamaaCaaabeqcfasaaiaadIhajuaGdaWgaaqcfasaaiaadMgaaeqa aaaaaKqbagaacaWGhbGaaiikaiabeI7aXjaacMcacqGHsislcaWGIb WaaSbaaKqbGeaacaaIWaaajuaGbeaaaaaacaGLOaGaayzkaaWaaWba aeqajuaibaGaamyyaKqbaoaaBaaajuaibaGaamyAaaqabaaaaaaa@502B@  …(2.10)

Now the conditional log likelihood function is given by

log L (θ; x _ )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGacYgaca GGVbGaai4zaiaadYeadaahaaqabKqbGeaacqGHxiIkaaqcfaOaaiik aiabeI7aXjaaykW7caGG7aWaaWaaaeaacaWG4baaaiaaysW7caGGPa GaaGjbVlabg2da9aaa@466D@ i=1 n n 0 log b x i + i=1 n n 0 x i log(θ) i=1 n n 0 log( G(θ) b 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaqahaba GaciiBaiaac+gacaGGNbGaamOyamaaBaaajuaibaGaamiEaKqbaoaa BaaajuaibaGaamyAaaqabaaajuaGbeaacqGHRaWkdaaeWbqaaiaadI hadaWgaaqcfasaaiaadMgaaKqbagqaaiGacYgacaGGVbGaai4zaiaa cIcacqaH4oqCcaGGPaaajuaibaGaamyAaiabg2da9iaaigdaaeaaca WGUbGaeyOeI0IaamOBaKqbaoaaBaaajuaibaGaaGimaaqabaaajuaG cqGHris5aiabgkHiTmaaqahabaGaciiBaiaac+gacaGGNbWaaeWaae aacaWGhbGaaiikaiabeI7aXjaacMcacqGHsislcaWGIbWaaSbaaKqb GeaacaaIWaaajuaGbeaaaiaawIcacaGLPaaaaKqbGeaacaWGPbGaey ypa0JaaGymaaqaaiaad6gacqGHsislcaWGUbqcfa4aaSbaaKqbGeaa caaIWaaabeaaaKqbakabggHiLdaajuaibaGaamyAaiabg2da9iaaig daaeaacaWGUbGaeyOeI0IaamOBaKqbaoaaBaaajuaibaGaaGimaaqa baaajuaGcqGHris5aaaa@71E7@  …(2.11)

The mle θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaiaaaaa@383E@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  is the solution to an equation

  x ¯ = θ ˜ G ( θ ˜ ) (G( θ ˜ ) b 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qeaiabg2da9maalaaabaGafqiUdeNbaGaacaaMi8Uabm4rayaafaGa aiikaiqbeI7aXzaaiaGaaiykaaqaaiaacIcacaWGhbGaaiikaiqbeI 7aXzaaiaGaaiykaiabgkHiTiaadkgadaWgaaqcfasaaiaaicdaaKqb agqaaiaacMcaaaaaaa@489E@  ,                                     …(2.12)

where x ¯ = i=1 n n o x i n n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qeaiabg2da9maalaaabaWaaabCaeaacaWG4bWaaSbaaKqbGeaacaWG PbaajuaGbeaaaKqbGeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gacq GHsislcaWGUbqcfa4aaSbaaKqbGeaacaWGVbaabeaaaKqbakabggHi LdaabaGaamOBaiabgkHiTiaad6gadaWgaaqcfasaaiaaicdaaKqbag qaaaaaaaa@4A0F@  is the mean of the positive observations only. We note that mle of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  based on full likelihood (eq. 2.6) and based on conditional likelihood (Eq. 2.12) are the same and

A V θ ˜ (θ)= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiabeI7a XjaacMcacqGH9aqpaaa@3ED1@ ( (n n 0 ){ ( G (θ) θ(G(θ) b 0 ) + G (θ)(G(θ) b 0 ) G (θ) 2 (G(θ) b 0 ) 2 ) } ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba Gaaiikaiaad6gacqGHsislcaWGUbWaaSbaaKqbGeaacaaIWaaajuaG beaacaGGPaWaaiWaaeaadaqadaqaamaalaaabaGaaGjcVlqadEeaga qbaiaacIcacqaH4oqCcaGGPaaabaGaeqiUdeNaaiikaiaadEeacaGG OaGaeqiUdeNaaiykaiabgkHiTiaadkgadaWgaaqcfasaaiaaicdaaK qbagqaaiaacMcaaaGaey4kaSYaaSaaaeaaceWGhbGbayaacaGGOaGa eqiUdeNaaiykaiaacIcacaWGhbGaaiikaiabeI7aXjaacMcacqGHsi slcaWGIbWaaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPaGaeyOeI0Ia bm4rayaafaGaaiikaiabeI7aXjaacMcadaahaaqabKqbGeaacaaIYa aaaaqcfayaaiaacIcacaWGhbGaaiikaiabeI7aXjaacMcacqGHsisl caWGIbWaaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPaWaaWbaaeqaju aibaGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaaacaGL7bGaayzFaaaa caGLOaGaayzkaaWaaWbaaeqajuaibaGaeyOeI0IaaGymaaaaaaa@7174@  …(2.13)

Assuming that Cramer-Huzurbazar conditions required for asymptotic normality for MLEs are satisfied, we have following theorem:

Theorem 2.2: Let X 1 , X 2 ,... X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada WgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWGybWaaSbaaKqbGeaa caaIYaaajuaGbeaacaGGSaGaaiOlaiaac6cacaGGUaGaamiwamaaBa aajuaibaGaamOBaaqcfayabaaaaa@4187@  be a random sample from ZITPSD with parameters π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ . Then the mle of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  is solution to the eq. (2.12) and has asymptotic normal distribution with mean θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  and variance A V θ ˜ (θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiabeI7a XjaacMcaaaa@3DCB@  for n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gaaa a@376C@  sufficiently large. That is as n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gacq GHsgIRcqGHEisPaaa@3ACA@ , ( n ( θ ˜ θ) ) N 1 (0,A V θ ˜ (θ)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaOaaaeaacaWGUbaabeaacaGGOaGafqiUdeNbaGaacqGHsislcqaH 4oqCcaGGPaaacaGLOaGaayzkaaGaeyOKH4QaamOtamaaBaaajuaiba GaaGymaaqcfayabaGaaiikaiaaicdacaGGSaGaaGPaVlaaykW7caWG bbGaamOvamaaBaaajuaibaGafqiUdeNbaGaaaKqbagqaaiaacIcacq aH4oqCcaGGPaGaaiykaaaa@5049@ .

In the following we present moment estimator of ZITPSD.

Moment estimator of ZITPSD: We have,

E(X)= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweaca GGOaGaamiwaiaacMcacqGH9aqpaaa@3A7F@ πθ G (θ) G(θ) =πμ(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeqiWdaNaaGPaVlabeI7aXjaayIW7ceWGhbGbauaacaGGOaGaeqiU deNaaiykaaqaaiaadEeacaGGOaGaeqiUdeNaaiykaaaacaaMe8Uaey ypa0JaeqiWdaNaaGPaVlabeY7aTjaacIcacqaH4oqCcaGGPaaaaa@4F7A@  

E( X 2 )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweaca GGOaGaamiwamaaCaaabeqcfasaaiaaikdaaaqcfaOaaiykaiabg2da 9aaa@3C19@ θπ G(θ) ( θ G (θ)+ G (θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeqiUdeNaaGjbVlabec8aWbqaaiaadEeacaGGOaGaeqiUdeNaaiyk aaaadaqadaqaaiabeI7aXjaaysW7ceWGhbGbayaacaGGOaGaeqiUde NaaiykaiaaysW7cqGHRaWkceWGhbGbauaacaGGOaGaeqiUdehacaGL OaGaayzkaaaaaa@4DC1@  and

Var(X)= θπ G(θ) ( θ G (θ)+ G (θ) πθ ( G (θ ) 2 G(θ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfaca WGHbGaamOCaiaacIcacaWGybGaaiykaiaaysW7cqGH9aqpdaWcaaqa aiabeI7aXjaaysW7cqaHapaCaeaacaWGhbGaaiikaiabeI7aXjaacM caaaWaaeWaaeaacqaH4oqCcaaMe8Uabm4rayaagaGaaiikaiabeI7a XjaacMcacaaMe8Uaey4kaSIabm4rayaafaGaaiikaiabeI7aXjaacM cacqGHsisldaWcaaqaaiaaysW7cqaHapaCcaaMe8UaeqiUdeNaaGPa VpaabmaabaGabm4rayaafaGaaiikaiabeI7aXbGaayjkaiaawMcaam aaCaaabeqcfasaaiaaikdaaaaajuaGbaGaam4raiaacIcacqaH4oqC caGGPaaaaaGaayjkaiaawMcaaaaa@693C@ ,

= σ 2 (π,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9i abeo8aZnaaCaaabeqcfasaaiaaikdaaaqcfaOaaiikaiabec8aWjaa cYcacqaH4oqCcaGGPaaaaa@4058@  say.

Let,

  X ¯ =πμ(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIfaga qeaiabg2da9iabec8aWjaaykW7cqaH8oqBcaGGOaGaeqiUdeNaaiyk aiaaysW7aaa@420E@  …(2.14)

i=1 n x i 2 n = πθ G(θ) ( θ G (θ)+ G (θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaabCaeaacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaadaahaaqa bKqbGeaacaaIYaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaa qcfaOaeyyeIuoaaKqbGeaacaWGUbaaaKqbakabg2da9maalaaabaGa eqiWdaNaaGPaVlabeI7aXbqaaiaadEeacaGGOaGaeqiUdeNaaiykaa aadaqadaqaaiabeI7aXjaaysW7ceWGhbGbayaacaGGOaGaeqiUdeNa aiykaiaaysW7cqGHRaWkceWGhbGbauaacaGGOaGaeqiUdehacaGLOa Gaayzkaaaaaa@5AB2@ , …(2.15)

Solving eq. (2.14) and eq. (2.15) we get moment estimators of π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ .

Theorem 2.3: Let X 1 , X 2 ,... X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada WgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWGybWaaSbaaKqbGeaa caaIYaaajuaGbeaacaGGSaGaaiOlaiaac6cacaGGUaGaamiwamaaBa aajuaibaGaamOBaaqcfayabaaaaa@4187@  be a random sample from ZITPSD with parameters π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ . Then the moment estimator of π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  are obtained by solving in the eq. (2.14) and eq. (2.15). The moment estimator of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  has asymptotic normal distribution with mean πμ(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWj aaykW7cqaH8oqBcaGGOaGaeqiUdeNaaiykaaaa@3E86@  and variance σ 2 (π,θ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaysW7da Wcaaqaaiabeo8aZnaaCaaabeqcfasaaiaaikdaaaqcfaOaaiikaiab ec8aWjaacYcacqaH4oqCcaGGPaaabaGaamOBaaaaaaa@41E2@ , for n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@  sufficiently large. That is as n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gacq GHsgIRcqGHEisPaaa@3ACA@ , ( n ( θ ¯ θ) ) N 1 ( 0, σ 2 (π,θ) n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaOaaaeaacaWGUbaabeaacaGGOaGafqiUdeNbaebacqGHsislcqaH 4oqCcaGGPaaacaGLOaGaayzkaaGaeyOKH4QaamOtamaaBaaajuaiba GaaGymaaqcfayabaWaaeWaaeaacaaIWaGaaiilaiaaykW7caaMc8+a aSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaKqbakaacIcacq aHapaCcaGGSaGaeqiUdeNaaiykaaqaaiaad6gaaaaacaGLOaGaayzk aaaaaa@530C@ .

Tests for the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  of ZITPS distribution

Test based on θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@ : Suppose we wish to test H 0 :θ= θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaiaacQdacqaH4oqCcqGH9aqpcqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaaa@3FA4@  vs H 1 :θ θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaigdaaKqbagqaaiaacQdacqaH4oqCcqGHGjsUcqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaaa@4066@ . Let us assume that π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  is known. Therefore, under H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada Wgaaqaaiaaicdaaeqaaaaa@3821@ , from Theorem (2.1) we have

( θ ^ θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GafqiUdeNbaKaacqGHsislcqaH4oqCdaWgaaqcfasaaiaaicdaaKqb agqaaaGaayjkaiaawMcaaaaa@3E02@ ~ AN( 0,A V θ ^ (π, θ 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGobWaaeWaaeaacaaIWaGaaiilaiaadgeacaWGwbWaaSbaaKqbGeaa cuaH4oqCgaqcaaqcfayabaGaaiikaiabec8aWjaacYcacqaH4oqCda WgaaqcfasaaiaaicdaaKqbagqaaiaacMcaaiaawIcacaGLPaaaaaa@465C@ .                                 …(2.16)

Define a test statistic to be Z 1 = θ ^ θ 0 A V θ ^ (π, θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaigdaaKqbagqaaiabg2da9maalaaabaGafqiUdeNb aKaacqGHsislcqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaqaam aakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaajaaajuaG beaacaGGOaGaeqiWdaNaaiilaiabeI7aXnaaBaaajuaibaGaaGimaa qcfayabaGaaiykaaqabaaaaaaa@4B6D@ . Based on Z 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38F0@ we define the test ψ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGymaaqcfayabaaaaa@39DF@  which rejects H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@  at α level of significance, if | Z 1 |> z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GaamOwamaaBaaajuaibaGaaGymaaqcfayabaaacaGLhWUaayjcSdGa eyOpa4JaamOEamaaBaaajuaibaGaaGymaiabgkHiTiabeg7aHjaac+ cacaaIYaaajuaGbeaaaaa@43AC@ , where z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaaIXaGaeyOeI0IaeqySdeMaai4laiaaikdaaeqaaaaa@3BD8@  is the upper 100(α/2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdaca aIWaGaaGimaiaacIcacqaHXoqycaGGVaGaaGOmaiaacMcaaaa@3D0F@ th percentile of SNV.

Let Φ(.) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfA6agj aacIcacaGGUaGaaiykaaaa@39FE@ be the cumulative distribution function of SNV. Then the power of the test ψ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGymaaqcfayabaaaaa@39DF@  is given by

β ψ 1 (π,θ)= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGaeqiYdKxcfa4aaSbaaKqbGeaacaaIXaaabeaaaKqb agqaaiaacIcacqaHapaCcaGGSaGaeqiUdeNaaiykaiabg2da9aaa@42DF@ 1Φ(B)+Φ(A) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHsislcqqHMoGrcaGGOaGaamOqaiaacMcacqGHRaWkcqqHMoGrcaGG OaGaamyqaiaacMcaaaa@4036@ ,                                 

where    

  A= θ 0 θ z 1α/2 A V θ ^ (π, θ 0 ) A V θ ^ (π,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeacq GH9aqpdaWcaaqaaiabeI7aXnaaBaaajuaibaGaaGimaaqcfayabaGa eyOeI0IaeqiUdeNaeyOeI0IaamOEamaaBaaajuaibaGaaGymaiabgk HiTiabeg7aHjaac+cacaaIYaaajuaGbeaadaGcaaqaaiaadgeacaWG wbWaaSbaaKqbGeaacuaH4oqCgaqcaaqcfayabaGaaiikaiabec8aWj aacYcacqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaqabaGaaiyk aaqaamaakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaaja aajuaGbeaacaGGOaGaeqiWdaNaaiilaiabeI7aXbqabaGaaiykaaaa aaa@5AFB@  and

B= θ 0 θ+ z 1α/2 A V θ ^ (π, θ 0 ) A V θ ^ (π,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkeacq GH9aqpdaWcaaqaaiabeI7aXnaaBaaajuaibaGaaGimaaqcfayabaGa eyOeI0IaeqiUdeNaey4kaSIaamOEamaaBaaajuaibaGaaGymaiabgk HiTiabeg7aHjaac+cacaaIYaaajuaGbeaadaGcaaqaaiaadgeacaWG wbWaaSbaaKqbGeaacuaH4oqCgaqcaaqcfayabaGaaiikaiabec8aWj aacYcacqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaqabaGaaiyk aaqaamaakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaaja aajuaGbeaacaGGOaGaeqiWdaNaaiilaiabeI7aXbqabaGaaiykaaaa aaa@5AF1@ .

However, in practice π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  is unknown. Hence we modify the test statistic by replacing π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  by its maximum likelihood estimator ( π ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaWaaSbaaKqbGeaacaaIWaaajuaGbeaaaaa@39DD@ ), when H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@  is true. By doing so, we define test Z 1 = θ ^ θ 0 A V θ ^ ( π ^ 0 , θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadQfaga qbamaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0ZaaSaaaeaacuaH 4oqCgaqcaiabgkHiTiabeI7aXnaaBaaajuaibaGaaGimaaqcfayaba aabaWaaOaaaeaacaWGbbGaamOvamaaBaaajuaibaGafqiUdeNbaKaa aKqbagqaaiaacIcacuaHapaCgaqcamaaBaaajuaibaGaaGimaaqcfa yabaGaaiilaiabeI7aXnaaBaaajuaibaGaaGimaaqcfayabaaabeaa caGGPaaaaaaa@4D20@ , where π ^ 0 = (n n 0 )f( θ 0 ) n(f( θ 0 ) b 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpdaWcaaqaaiaa cIcacaWGUbGaeyOeI0IaamOBamaaBaaajuaibaGaaGimaaqcfayaba GaaiykaiaadAgacaGGOaGaeqiUde3aaSbaaKqbGeaacaaIWaaajuaG beaacaGGPaaabaGaamOBaiaaysW7caGGOaGaamOzaiaacIcacqaH4o qCdaWgaaqcfasaaiaaicdaaKqbagqaaiaacMcacqGHsislcaWGIbWa aSbaaKqbGeaacaaIWaaajuaGbeaacaGGPaaaaaaa@531C@

Based on Z 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadQfaga qbamaaBaaajuaibaGaaGymaaqcfayabaaaaa@38FC@ , we propose a test ψ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI8a5z aafaWaaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@39EB@  rejects H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@  at α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHb aa@3818@  level of significance, if | Z 1 |> Z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GabmOwayaafaWaaSbaaKqbGeaacaaIXaaajuaGbeaaaiaawEa7caGL iWoacqGH+aGpcaWGAbWaaSbaaKqbGeaacaaIXaGaeyOeI0IaeqySde Maai4laiaaikdaaKqbagqaaaaa@4398@ .

The power of this test is given by

β ψ 1 (π,θ)= k=0 n ( 1Φ( B ^ k )+Φ( A ^ k ) )P( n 0 =k) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGafqiYdKNbauaajuaGdaWgaaqcfasaaiaaigdaaeqa aaqcfayabaGaaiikaiabec8aWjaacYcacqaH4oqCcaGGPaGaeyypa0 ZaaabCaeaadaqadaqaaiaaigdacqGHsislcqqHMoGrcaGGOaGabmOq ayaajaWaaSbaaKqbGeaacaWGRbaajuaGbeaacaGGPaGaey4kaSIaeu OPdyKaaiikaiqadgeagaqcamaaBaaajuaibaGaam4AaaqcfayabaGa aiykaaGaayjkaiaawMcaaiaadcfacaGGOaGaamOBamaaBaaajuaiba GaaGimaaqcfayabaGaeyypa0Jaam4AaiaacMcaaKqbGeaacaWGRbGa eyypa0JaaGimaaqaaiaad6gaaKqbakabggHiLdaaaa@5F22@ , ...(2.17)

where

B ^ k = θ 0 θ+ z 1α/2 A V θ ^ ( π ^ 0 , θ 0 ) A V θ ^ (π,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkeaga qcamaaBaaajuaibaGaam4AaaqcfayabaGaeyypa0ZaaSaaaeaacqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaiabgkHiTiabeI7aXjabgU caRiaadQhadaWgaaqcfasaaiaaigdacqGHsislcqaHXoqycaGGVaGa aGOmaaqcfayabaWaaOaaaeaacaWGbbGaamOvamaaBaaajuaibaGafq iUdeNbaKaaaKqbagqaaiaacIcacuaHapaCgaqcamaaBaaajuaibaGa aGimaaqcfayabaGaaiilaiabeI7aXnaaBaaajuaibaGaaGimaaqcfa yabaaabeaacaGGPaaabaWaaOaaaeaacaWGbbGaamOvamaaBaaajuai baGafqiUdeNbaKaaaKqbagqaaiaacIcacqaHapaCcaGGSaGaeqiUde habeaacaGGPaaaaaaa@5E75@  , A ^ k = θ 0 θ z 1α/2 A V θ ^ ( π ^ 0 , θ 0 ) A V θ ^ (π,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadgeaga qcamaaBaaajuaibaGaam4AaaqcfayabaGaeyypa0ZaaSaaaeaacqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaiabgkHiTiabeI7aXjabgk HiTiaadQhadaWgaaqcfasaaiaaigdacqGHsislcqaHXoqycaGGVaGa aGOmaaqcfayabaWaaOaaaeaacaWGbbGaamOvamaaBaaajuaibaGafq iUdeNbaKaaaKqbagqaaiaacIcacuaHapaCgaqcamaaBaaajuaibaGa aGimaaqcfayabaGaaiilaiabeI7aXnaaBaaajuaibaGaaGimaaqcfa yabaaabeaacaGGPaaabaWaaOaaaeaacaWGbbGaamOvamaaBaaajuai baGafqiUdeNbaKaaaKqbagqaaiaacIcacqaHapaCcaGGSaGaeqiUde habeaacaGGPaaaaaaa@5E7F@ ,

P( n 0 =k)=( k n ) P 0 k (1 P 0 ) nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca GGOaGaamOBamaaBaaajuaibaGaaGimaaqcfayabaGaeyypa0Jaam4A aiaacMcacqGH9aqpdaqadaqaamaaDeaabaGaam4Aaaqaaiaad6gaaa aacaGLOaGaayzkaaGaamiuamaaBaaajuaibaGaaGimaaqcfayabaWa aWbaaeqajuaibaGaam4AaaaajuaGcaGGOaGaaGymaiabgkHiTiaadc fadaWgaaqcfasaaiaaicdaaKqbagqaaiaacMcadaahaaqabKqbGeaa caWGUbGaeyOeI0Iaam4AaaaajuaGcaaMc8UaaGPaVdaa@5227@ ,with P 0 =( 1π+π b 0 f(θ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada WgaaqcfasaaiaaicdaaKqbagqaaiabg2da9maabmaabaGaaGymaiab gkHiTiabec8aWjabgUcaRiabec8aWnaalaaabaGaamOyamaaBaaaju aibaGaaGimaaqcfayabaaabaGaamOzaiaacIcacqaH4oqCcaGGPaaa aaGaayjkaiaawMcaaaaa@4800@ . …(2.18)

Below we develop test based on θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaiaaaaa@383E@ , estimator based on conditional likelihood approach.

Test based on θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaiaaaaa@383E@ : Theorem (2.5) gives

  ( θ ˜ θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GafqiUdeNbaGaacqGHsislcqaH4oqCdaWgaaqcfasaaiaaicdaaKqb agqaaaGaayjkaiaawMcaaaaa@3E01@ ~ AN( 0,A V θ ˜ ( θ 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGobWaaeWaaeaacaaIWaGaaiilaiaadgeacaWGwbWaaSbaaKqbGeaa cuaH4oqCgaacaaqcfayabaGaaiikaiabeI7aXnaaBaaajuaibaGaaG imaaqcfayabaGaaiykaaGaayjkaiaawMcaaaaa@43EE@ . …(2.19)

Hence, we define test statistic Z 2 = θ ˜ θ 0 A V θ ˜ ( θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaikdaaKqbagqaaiabg2da9maalaaabaGafqiUdeNb aGaacqGHsislcqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaqaam aakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaaiaaajuaG beaacaGGOaGaeqiUde3aaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPa aabeaaaaaaaa@48FF@ . A test based on Z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38F1@  which rejects H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@  α level of significance, if | Z 2 |> z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GaaGPaVlaadQfadaWgaaqcfasaaiaaikdaaKqbagqaaaGaay5bSlaa wIa7aiabg6da+iaadQhadaWgaaqcfasaaiaaigdacqGHsislcqaHXo qycaGGVaGaaGOmaaqcfayabaaaaa@4538@ .

The power of the test ψ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGOmaaqcfayabaaaaa@39E0@  is given by

  β ψ 2 (π,θ)=1Φ(B)+Φ(A) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGaeqiYdKxcfa4aaSbaaKqbGeaacaaIYaaabeaaaKqb agqaaiaacIcacqaHapaCcaGGSaGaeqiUdeNaaiykaiabg2da9iaaig dacqGHsislcqqHMoGrcaGGOaGaamOqaiaacMcacqGHRaWkcqqHMoGr caGGOaGaamyqaiaacMcaaaa@4C9D@  , …(2.20)

where, A= θ 0 θ z 1α/2 A V θ ˜ ( θ 0 ) A V θ ^ (θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeacq GH9aqpdaWcaaqaaiabeI7aXnaaBaaajuaibaGaaGimaaqcfayabaGa eyOeI0IaeqiUdeNaeyOeI0IaamOEamaaBaaajuaibaGaaGymaiabgk HiTiabeg7aHjaac+cacaaIYaaajuaGbeaadaGcaaqaaiaadgeacaWG wbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiabeI7aXn aaBaaajuaibaGaaGimaaqcfayabaGaaiykaaqabaaabaWaaOaaaeaa caWGbbGaamOvamaaBaaajuaibaGafqiUdeNbaKaaaKqbagqaaiaacI cacqaH4oqCaeqaaiaacMcaaaaaaa@5620@ , B= θ 0 θ+ z 1α/2 A V θ ˜ ( θ 0 ) A V θ ^ (θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkeacq GH9aqpdaWcaaqaaiabeI7aXnaaBaaajuaibaGaaGimaaqcfayabaGa eyOeI0IaeqiUdeNaey4kaSIaamOEamaaBaaajuaibaGaaGymaiabgk HiTiabeg7aHjaac+cacaaIYaaajuaGbeaadaGcaaqaaiaadgeacaWG wbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiabeI7aXn aaBaaajuaibaGaaGimaaqcfayabaGaaiykaaqabaaabaWaaOaaaeaa caWGbbGaamOvamaaBaaajuaibaGafqiUdeNbaKaaaKqbagqaaiaacI cacqaH4oqCaeqaaiaacMcaaaaaaa@5616@ .      

Test based on the moment estimator θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaraaaaa@3847@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ : It is clear that the problem of testing H 0 :θ= θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaiaacQdacqaH4oqCcqGH9aqpcqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaaa@3FA4@  vs H 1 :θ θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaigdaaKqbagqaaiaacQdacqaH4oqCcqGHGjsUcqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaaa@4066@  is equivalent to testing H 0 :μ(θ)=μ( θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaiaacQdacqaH8oqBcaGGOaGaeqiU deNaaiykaiabg2da9iabeY7aTjaacIcacqaH4oqCdaWgaaqcfasaai aaicdaaKqbagqaaiaacMcaaaa@45C2@  vs H 1 :μ(θ)μ( θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaigdaaKqbagqaaiaacQdacqaH8oqBcaGGOaGaeqiU deNaaiykaiabgcMi5kabeY7aTjaacIcacqaH4oqCdaWgaaqcfasaai aaicdaaKqbagqaaiaacMcaaaa@4684@ , where μ(θ)= θ G (θ) G(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeY7aTj aacIcacqaH4oqCcaGGPaGaeyypa0ZaaSaaaeaacaaMc8UaaGjbVlab eI7aXjaayIW7ceWGhbGbauaacaGGOaGaeqiUdeNaaGjbVlaacMcaae aacaWGhbGaaiikaiabeI7aXjaacMcaaaaaaa@4C02@ . We have from Theorem (2.3), sample mean is consistent and asymptotically normal for the population mean.

That is X ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIfaga qeaaaa@376E@ ~ AN( πμ(θ), σ 2 (π,θ) n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGobGaaGjbVpaabmaabaGaeqiWdaNaaGPaVlabeY7aTjaacIcacqaH 4oqCcaaMc8UaaiykaiaacYcacaaMc8+aaSaaaeaacqaHdpWCdaahaa qabKqbGeaacaaIYaaaaKqbakaacIcacqaHapaCcaGGSaGaeqiUdeNa aiykaaqaaiaad6gaaaaacaGLOaGaayzkaaaaaa@50D7@ .

Therefore, under H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@ , we have

n ( X ¯ π μ( θ 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaakaaaba GaamOBaiaaysW7aeqaamaabmaabaWaaSaaaeaaceWGybGbaebaaeaa cqaHapaCaaGaeyOeI0IaeqiVd0MaaiikaiabeI7aXnaaBaaajuaiba GaaGimaaqcfayabaGaaiykaaGaayjkaiaawMcaaaaa@449D@ ~ AN( 0, σ 2 (π, θ 0 ) π 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGobGaaGjbVpaabmaabaGaaGimaiaacYcadaWcaaqaaiabeo8aZnaa CaaabeqcfasaaiaaikdaaaqcfaOaaiikaiabec8aWjaacYcacqaH4o qCdaWgaaqcfasaaiaaicdaaKqbagqaaiaacMcaaeaacqaHapaCdaah aaqabKqbGeaacaaIYaaaaaaaaKqbakaawIcacaGLPaaaaaa@4A69@ .

Define test statistic

Z 3 = n ( X ¯ π μ( θ 0 ) ) σ 2 (π, θ 0 ) π 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaiodaaKqbagqaaiabg2da9maalaaabaWaaOaaaeaa caWGUbGaaGjbVdqabaWaaeWaaeaadaWcaaqaaiqadIfagaqeaaqaai abec8aWbaacqGHsislcqaH8oqBcaGGOaGaeqiUde3aaSbaaKqbGeaa caaIWaaajuaGbeaacaGGPaaacaGLOaGaayzkaaaabaWaaOaaaeaada Wcaaqaaiabeo8aZnaaCaaabeqcfasaaiaaikdaaaqcfaOaaiikaiab ec8aWjaacYcacqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaiaacM caaeaacqaHapaCdaahaaqabKqbGeaacaaIYaaaaaaaaKqbagqaaaaa aaa@5613@ ~ N( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6eaca aMe8+aaeWaaeaacaaIWaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa@3C87@ , when π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  is known.

The test ψ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaG4maaqcfayabaaaaa@39E1@  rejects H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@  at α level of significance if | Z 3 |> z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GaamOwamaaBaaajuaibaGaaG4maaqcfayabaaacaGLhWUaayjcSdGa eyOpa4JaamOEamaaBaaajuaibaGaaGymaiabgkHiTiabeg7aHjaac+ cacaaIYaaajuaGbeaaaaa@43AE@ .

That is, reject H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@ if ( n | X ¯ π μ( θ 0 ) | σ 2 (π, θ 0 ) π 2 )> z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaSaaaeaadaGcaaqaaiaad6gacaaMe8oabeaadaabdaqaamaalaaa baGabmiwayaaraaabaGaeqiWdahaaiabgkHiTiabeY7aTjaacIcacq aH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaiaacMcaaiaawEa7caGL iWoaaeaadaGcaaqaamaalaaabaGaeq4Wdm3aaWbaaeqajuaibaGaaG OmaaaajuaGcaGGOaGaeqiWdaNaaiilaiabeI7aXnaaBaaajuaibaGa aGimaaqcfayabaGaaiykaaqaaiabec8aWnaaCaaabeqcfasaaiaaik daaaaaaaqcfayabaaaaaGaayjkaiaawMcaaiabg6da+iaadQhadaWg aaqcfasaaiaaigdacqGHsislcqaHXoqycaGGVaGaaGOmaaqcfayaba aaaa@5D50@ .

The power of the test ψ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaG4maaqcfayabaaaaa@39E1@ is given by

β ψ 3 (π,θ)= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGaeqiYdKxcfa4aaSbaaKqbGeaacaaIZaaabeaaaKqb agqaaiaacIcacqaHapaCcaGGSaGaeqiUdeNaaiykaiabg2da9aaa@42E1@ 1Φ( B )+Φ( A ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHsislcqqHMoGrcaGGOaGabmOqayaafaGaaiykaiabgUcaRiabfA6a gjaacIcaceWGbbGbauaacaGGPaaaaa@404E@ , …(2.21)

where A = π( μ( θ 0 ) z 1α/2 σ 2 (π, θ 0 ) n π 2 )πμ(θ) σ 2 (π,θ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadgeaga qbaiabg2da9maalaaabaGaeqiWda3aaeWaaeaacqaH8oqBcaGGOaGa eqiUde3aaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPaGaeyOeI0Iaam OEamaaBaaajuaibaGaaGymaiabgkHiTiabeg7aHjaac+cacaaIYaaa juaGbeaadaGcaaqaamaalaaabaGaeq4Wdm3aaWbaaeqajuaibaGaaG OmaaaajuaGcaGGOaGaeqiWdaNaaiilaiabeI7aXnaaBaaajuaibaGa aGimaaqcfayabaGaaiykaaqaaiaad6gacqaHapaCdaahaaqabKqbGe aacaaIYaaaaaaaaKqbagqaaaGaayjkaiaawMcaaiabgkHiTiabec8a WjaaykW7cqaH8oqBcaGGOaGaeqiUdeNaaiykaaqaamaakaaabaWaaS aaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaKqbakaacIcacqaH apaCcaGGSaGaeqiUdeNaaiykaaqaaiaad6gaaaaabeaaaaaaaa@6B42@  and

B = π( μ( θ 0 )+ z 1α/2 σ 2 (π, θ 0 ) n π 2 )πμ(θ) σ 2 (π,θ) n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkeaga qbaiabg2da9maalaaabaGaeqiWda3aaeWaaeaacqaH8oqBcaGGOaGa eqiUde3aaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPaGaey4kaSIaam OEamaaBaaajuaibaGaaGymaiabgkHiTiabeg7aHjaac+cacaaIYaaa juaGbeaadaGcaaqaamaalaaabaGaeq4Wdm3aaWbaaeqajuaibaGaaG OmaaaajuaGcaGGOaGaeqiWdaNaaiilaiabeI7aXnaaBaaajuaibaGa aGimaaqcfayabaGaaiykaaqaaiaad6gacqaHapaCdaahaaqabKqbGe aacaaIYaaaaaaaaKqbagqaaaGaayjkaiaawMcaaiabgkHiTiabec8a WjaaykW7cqaH8oqBcaGGOaGaeqiUdeNaaiykaaqaamaakaaabaWaaS aaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaKqbakaacIcacqaH apaCcaGGSaGaeqiUdeNaaiykaaqaaiaad6gaaaaabeaaaaGaaiilaa aa@6BE8@  

If π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  is unknown, we modify the test statistic by replacing π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  by its estimate ( π ^ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcacu aHapaCgaqcamaaBaaajuaibaGaaGimaaqcfayabaGaaiykaaaa@3B36@  under H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@ . By doing so, we define test statistic

Z 3 = n ( X ¯ π ^ 0 μ( θ 0 ) ) σ 2 ( π ^ 0 , θ 0 ) π ^ 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadQfaga qbamaaBaaajuaibaGaaG4maaqcfayabaGaeyypa0ZaaSaaaeaadaGc aaqaaiaad6gacaaMe8oabeaadaqadaqaamaalaaabaGabmiwayaara aabaGafqiWdaNbaKaadaWgaaqcfasaaiaaicdaaKqbagqaaaaacqGH sislcqaH8oqBcaGGOaGaeqiUde3aaSbaaKqbGeaacaaIWaaajuaGbe aacaGGPaaacaGLOaGaayzkaaaabaWaaOaaaeaadaWcaaqaaiabeo8a ZnaaCaaabeqcfasaaiaaikdaaaqcfaOaaiikaiqbec8aWzaajaWaaS baaKqbGeaacaaIWaaajuaGbeaacaGGSaGaeqiUde3aaSbaaKqbGeaa caaIWaaajuaGbeaacaGGPaaabaGafqiWdaNbaKaadaWgaaqcfasaai aaicdaaKqbagqaamaaCaaabeqcfasaaiaaikdaaaaaaaqcfayabaaa aaaa@5B14@ , …(2.22)

where π ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaWaaSbaaKqbGeaacaaIWaaajuaGbeaaaaa@39DD@  is given by π ^ 0 = X ¯ μ( θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpdaWcaaqaaiqa dIfagaqeaaqaaiabeY7aTjaaykW7caGGOaGaeqiUde3aaSbaaKqbGe aacaaIWaaajuaGbeaacaGGPaaaaaaa@43CF@ .

Based on Z 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadQfaga qbamaaBaaajuaibaGaaG4maaqcfayabaaaaa@38FE@  we propose a test ψ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI8a5z aafaWaaSbaaKqbGeaacaaIZaaajuaGbeaaaaa@39ED@  which rejects H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@  at α level of significance if | Z 3 |> z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GabmOwayaafaWaaSbaaKqbGeaacaaIZaaajuaGbeaaaiaawEa7caGL iWoacqGH+aGpcaWG6bWaaSbaaKqbGeaacaaIXaGaeyOeI0IaeqySde Maai4laiaaikdaaKqbagqaaaaa@43BA@ .

The power of the test is given by

β ψ 3 (π,θ)= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGafqiYdKNbauaajuaGdaWgaaqcfasaaiaaiodaaeqa aaqcfayabaGaaiikaiabec8aWjaacYcacqaH4oqCcaGGPaGaeyypa0 daaa@42ED@ k=0 n ( 1Φ( B k )+Φ( A k ) )P( n 0 =k) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaqahaba WaaeWaaeaacaaIXaGaeyOeI0IaeuOPdyKaaiikaiqadkeagaGbamaa BaaajuaibaGaam4AaaqcfayabaGaaiykaiabgUcaRiabfA6agjaacI caceWGbbGbayaadaWgaaqcfasaaiaadUgaaKqbagqaaiaacMcaaiaa wIcacaGLPaaacaWGqbGaaiikaiaad6gadaWgaaqcfasaaiaaicdaaK qbagqaaiabg2da9iaadUgacaGGPaaajuaibaGaam4Aaiabg2da9iaa icdaaKqbagaacaWGUbaacqGHris5aaaa@52AA@ , …(2.23)

where

A k = π ^ 0 ( μ( θ 0 ) z 1α/2 σ 2 ( π ^ 0 , θ 0 ) n π 2 )-πμ(θ) σ 2 (π,θ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmyqay aagaWaaSbaaKqbGeaacaWGRbaajuaGbeaacqGH9aqpdaWcaaqaaiqb ec8aWzaajaWaaSbaaKqbGeaacaaIWaaajuaGbeaadaqadaqaaiabeY 7aTjaacIcacqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaiaacMca cqGHsislcaWG6bWaaSbaaKqbGeaacaaIXaGaaGPaVlaaysW7cqGHsi slcaaMc8UaeqySdeMaai4laiaaikdaaKqbagqaamaakaaabaWaaSaa aeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaKqbakaacIcacuaHap aCgaqcamaaBaaajuaibaGaaGimaaqcfayabaGaaiilaiabeI7aXnaa BaaajuaibaGaaGimaaqcfayabaGaaiykaaqaaiaad6gacqaHapaCda ahaaqabKqbGeaacaaIYaaaaaaaaKqbagqaaaGaayjkaiaawMcaaGqa aiaa=1cacqaHapaCcaaMc8UaeqiVd0MaaiikaiabeI7aXjaacMcaae aadaGcaaqaamaalaaabaGaeq4Wdm3aaWbaaeqajuaibaGaaGOmaaaa juaGcaGGOaGaeqiWdaNaaiilaiabeI7aXjaacMcaaeaacaWGUbaaaa qabaaaaaaa@74D6@  

B k = π ^ 0 ( μ( θ 0 )+ z 1α/2 σ 2 ( π ^ 0 , θ 0 ) n π 2 )-πμ(θ) σ 2 (π,θ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmOqay aagaWaaSbaaKqbGeaacaWGRbaajuaGbeaacqGH9aqpdaWcaaqaaiqb ec8aWzaajaWaaSbaaKqbGeaacaaIWaaajuaGbeaadaqadaqaaiabeY 7aTjaacIcacqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaiaacMca cqGHRaWkcaWG6bWaaSbaaKqbGeaacaaIXaGaaGPaVlabgkHiTiabeg 7aHjaac+cacaaIYaaajuaGbeaadaGcaaqaamaalaaabaGaeq4Wdm3a aWbaaeqajuaibaGaaGOmaaaajuaGcaGGOaGafqiWdaNbaKaadaWgaa qcfasaaiaaicdaaKqbagqaaiaacYcacqaH4oqCdaWgaaqcfasaaiaa icdaaKqbagqaaiaacMcaaeaacaWGUbGaeqiWda3aaWbaaeqajuaiba GaaGOmaaaaaaaajuaGbeaaaiaawIcacaGLPaaaieaacaWFTaGaeqiW daNaaGPaVlabeY7aTjaacIcacqaH4oqCcaGGPaaabaWaaOaaaeaada Wcaaqaaiabeo8aZnaaCaaabeqcfasaaiaaikdaaaqcfaOaaiikaiab ec8aWjaacYcacqaH4oqCcaGGPaaabaGaamOBaaaaaeqaaaaaaaa@71B4@

and P( n 0 =k)=( k n ) P 0 k (1 P 0 ) nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca GGOaGaamOBamaaBaaajuaibaGaaGimaaqcfayabaGaeyypa0Jaam4A aiaacMcacqGH9aqpdaqadaqaamaaDeaabaGaam4Aaaqaaiaad6gaaa aacaGLOaGaayzkaaGaamiuamaaBaaajuaibaGaaGimaaqcfayabaWa aWbaaeqajuaibaGaam4AaaaajuaGcaGGOaGaaGymaiabgkHiTiaadc fadaWgaaqcfasaaiaaicdaaKqbagqaaiaacMcadaahaaqabKqbGeaa caWGUbGaeyOeI0Iaam4AaaaajuaGcaaMc8UaaGPaVdaa@5227@ , with P 0 =( 1π+π a 0 f(θ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada WgaaqcfasaaiaaicdaaKqbagqaaiabg2da9maabmaabaGaaGymaiab gkHiTiabec8aWjabgUcaRiabec8aWnaalaaabaGaamyyamaaBaaaju aibaGaaGimaaqcfayabaaabaGaamOzaiaacIcacqaH4oqCcaGGPaaa aaGaayjkaiaawMcaaaaa@47FF@ .

Using the tests developed above, we can define two sided asymptotic confidence intervals for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ , by inverting acceptance regions of the tests appropriately. Below we report the same.

Asymptotic confidence interval for the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@

Asymptotic confidence interval for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  based on the test ψ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGymaaqcfayabaaaaa@39DF@ is given by

( θ ^ z 1α/2 A V θ ^ ( π ^ , θ ^ ) , θ ^ + z 1α/2 A V θ ^ ( π ^ , θ ^ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GafqiUdeNbaKaacqGHsislcaWG6bWaaSbaaKqbGeaacaaIXaGaeyOe I0IaeqySdeMaai4laiaaikdaaKqbagqaamaakaaabaGaamyqaiaadA fadaWgaaqcfasaaiqbeI7aXzaajaaajuaGbeaacaGGOaGafqiWdaNb aKaacaGGSaGafqiUdeNbaKaacaGGPaaabeaacaGGSaGaaGPaVlaayw W7cuaH4oqCgaqcaiabgUcaRiaadQhadaWgaaqcfasaaiaaigdacqGH sislcqaHXoqycaGGVaGaaGOmaaqcfayabaWaaOaaaeaacaWGbbGaam OvamaaBaaajuaibaGafqiUdeNbaKaaaKqbagqaaiaacIcacuaHapaC gaqcaiaacYcacuaH4oqCgaqcaiaacMcaaeqaaaGaayjkaiaawMcaaa aa@622A@  …(2.24)

where, A V θ ^ ( π ^ , θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaqcaaqcfayabaGaaiikaiqbec8a WzaajaGaaiilaiqbeI7aXzaajaGaaiykaaaa@4059@  is an estimate of asymptotic variance of θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@  and asymptotic confidence interval for θ based on the test ψ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGOmaaqcfayabaaaaa@39E0@  is given by

  ( θ ˜ z 1α/2 A V θ ˜ ( θ ˜ ) , θ ˜ + z 1α/2 A V θ ˜ ( θ ˜ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GafqiUdeNbaGaacqGHsislcaWG6bWaaSbaaKqbGeaacaaIXaGaeyOe I0IaeqySdeMaai4laiaaikdaaKqbagqaamaakaaabaGaamyqaiaadA fadaWgaaqcfasaaiqbeI7aXzaaiaaajuaGbeaacaGGOaGafqiUdeNb aGaacaGGPaaabeaacaGGSaGaaGPaVlaaywW7cuaH4oqCgaacaiabgU caRiaadQhadaWgaaqcfasaaiaaigdacqGHsislcqaHXoqycaGGVaGa aGOmaaqcfayabaWaaOaaaeaacaWGbbGaamOvamaaBaaajuaibaGafq iUdeNbaGaaaKqbagqaaiaacIcacuaH4oqCgaacaiaacMcaaeqaaaGa ayjkaiaawMcaaaaa@5D2A@  …(2.25)

where A V θ ˜ ( θ ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiqbeI7a XzaaiaGaaiykaaaa@3DDA@  is an estimate of the asymptotic variance of θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaiaaaaa@383E@  as given in the eq. (2.13) .

Asymptotic confidence interval for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  based on the test ψ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaG4maaqcfayabaaaaa@39E1@  is given by

( X ¯ π ^ z 1α/2 A V θ ¯ ( π ^ , θ ¯ ) , X ¯ π ^ + z 1α/2 A V θ ¯ ( π ^ , θ ¯ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaSaaaeaaceWGybGbaebaaeaacuaHapaCgaqcaaaacqGHsislcaWG 6bWaaSbaaKqbGeaacaaIXaGaeyOeI0IaeqySdeMaai4laiaaikdaaK qbagqaamaakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaa raaajuaGbeaacaGGOaGafqiWdaNbaKaacaGGSaGafqiUdeNbaebaca GGPaaabeaacaGGSaGaaGPaVlaaywW7daWcaaqaaiqadIfagaqeaaqa aiqbec8aWzaajaaaaiabgUcaRiaadQhadaWgaaqcfasaaiaaigdacq GHsislcqaHXoqycaGGVaGaaGOmaaqcfayabaWaaOaaaeaacaWGbbGa amOvamaaBaaajuaibaGafqiUdeNbaebaaKqbagqaaiaacIcacuaHap aCgaqcaiaacYcacuaH4oqCgaqeaiaacMcaaeqaaaGaayjkaiaawMca aaaa@6462@ , …(2.26)

where A V θ ¯ ( π ^ , θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaqeaaqcfayabaGaaiikaiqbec8a WzaajaGaaiilaiqbeI7aXzaaraGaaiykaaaa@4069@  = n ( X ¯ π μ( θ ) ) σ 2 (π,θ) π 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaOaaaeaacaWGUbGaaGjbVdqabaWaaeWaaeaadaWcaaqaaiqadIfa gaqeaaqaaiabec8aWbaacqGHsislcqaH8oqBcaGGOaGaeqiUde3aaS baaeaaaeqaaiaacMcaaiaawIcacaGLPaaaaeaadaGcaaqaamaalaaa baGaeq4Wdm3aaWbaaeqajuaibaGaaGOmaaaajuaGcaGGOaGaeqiWda NaaiilaiabeI7aXjaacMcaaeaacqaHapaCdaahaaqabKqbGeaacaaI YaaaaaaaaKqbagqaaaaaaaa@4F87@ .

In the following we study inference for zero-inflated Truncated Poisson distribution using results reported in the earlier.

Zero-Inflated Truncated Poisson Distribution

Truncated samples from discrete distributions arise in numerous situations where counts of zero are not observed. As an example, consider the distribution of the number of children per family in developing nations, where records are maintained only if there is at least a child in the family. The number of childless families remains unknown. The resulting sample is thus truncated with zero class missing. In continuous distribution, a sample of this type would be described as singly left truncated. In other situations, sample from discrete distributions might be censored on the right.

In this section, we consider zero-inflated truncated Poisson distribution truncated at right at the support point 't' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacEcaca WG0bGaai4jaaaa@38C8@  onwards, where 't' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacEcaca WG0bGaai4jaaaa@38C8@  is known. Moments, maximum likelihood estimators, Fisher information matrix for full and conditional likelihood are provided. We provide three tests for testing the parameter of the ZITPD.

Consider the probability mass function of truncated Poisson distribution (TPD) truncated at the support point 't' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacEcaca WG0bGaai4jaaaa@38C8@  onwards. The probability mass function of TPD is given by

P(X=x)= e θ θ x x!(1P(X>t) ,forx=0,1,2,...,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca GGOaGaamiwaiabg2da9iaadIhacaGGPaGaeyypa0ZaaSaaaeaacaWG LbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdehaaKqbakabeI7aXnaaCa aabeqcfasaaiaadIhaaaaajuaGbaGaamiEaiaaykW7caGGHaGaaiik aiaaigdacqGHsislcaWGqbGaaiikaiaadIfacqGH+aGpcaWG0bGaai ykaaaacaGGSaGaaGzbVlaadAgacaWGVbGaamOCaiaaysW7caWG4bGa eyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikdacaGGSaGaaiOlai aac6cacaGGUaGaaiilaiaadshaaaa@5FCA@

= e θ θ x x!( y=0 t e θ θ y y! ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXbaajuaG cqaH4oqCdaahaaqabKqbGeaacaWG4baaaaqcfayaaiaadIhacaaMc8 UaaiyiamaabmaabaWaaabCaeaadaWcaaqaaiaadwgadaahaaqabKqb GeaacqGHsislcqaH4oqCaaqcfaOaeqiUde3aaWbaaeqajuaibaGaam yEaaaaaKqbagaacaWG5bGaaGPaVlaacgcaaaaajuaibaGaamyEaiab g2da9iaaicdaaeaacaWG0baajuaGcqGHris5aaGaayjkaiaawMcaaa aacaGGSaGaaGzbVdaa@5857@                               

= θ x x!A(θ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaeqiUde3aaWbaaeqajuaibaGaamiEaaaaaKqbagaacaWG 4bGaaGPaVlaacgcacaWGbbGaaiikaiabeI7aXjaacMcaaaGaaiilai aaywW7aaa@4460@  where A(θ)=( y=0 t θ y y! ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca GGOaGaeqiUdeNaaiykaiabg2da9maabmaabaWaaabCaeaadaWcaaqa aiabeI7aXnaaCaaabeqcfasaaiaadMhaaaaajuaGbaGaamyEaiaayk W7caGGHaaaaaqcfasaaiaadMhacqGH9aqpcaaIWaaabaGaamiDaaqc faOaeyyeIuoaaiaawIcacaGLPaaaaaa@4A4A@  

Using this truncated distribution, we define the zero-inflated truncated Poisson distribution truncated at 't' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4jaiaads hacaaMc8Uaai4jaaaa@39D1@  onwards.

The probability mass function of ZITP distribution is given by

P(X=x)={ (1π)+ π A(θ)       for  x=0 π θ x x!A(θ)     for   x=1,2,3,...,t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca GGOaGaamiwaiabg2da9iaadIhacaGGPaGaeyypa0Zaaiqaaqaabeqa aiaacIcacaaIXaGaeyOeI0IaeqiWdaNaaiykaiabgUcaRmaalaaaba GaeqiWdahabaGaamyqaiaacIcacqaH4oqCcaGGPaaaaiaaysW7qaaa aaaaaaWdbiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOa8aaca WGMbGaam4BaiaadkhapeGaaiiOaiaacckapaGaamiEaiabg2da9iaa icdaaeaadaWcaaqaaiabec8aWjaaysW7cqaH4oqCdaahaaqabeaaca WG4baaaaqaaiaadIhacaaMc8UaaiyiaiaadgeacaGGOaGaeqiUdeNa aiykaaaapeGaaiiOaiaacckacaGGGcGaaiiOa8aacaWGMbGaam4Bai aadkhapeGaaiiOaiaacckacaGGGcWdaiaadIhacqGH9aqpcaaIXaGa aiilaiaaikdacaGGSaGaaG4maiaacYcacaGGUaGaaiOlaiaac6caca GGSaGaamiDaaaacaGL7baaaaa@7BC5@ and θ>0,0<π<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj abg6da+iaaicdacaaMc8UaaiilaiaaysW7caaIWaGaeyipaWJaeqiW daNaeyipaWJaaGymaaaa@42F3@  …(3.1)

Estimation of the parameters π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  

Estimation of the parameters using full likelihood function

Let X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada WgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaaMc8UaaGjbVlaadIfa daWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaaMe8UaaiOlaiaac6 cacaGGUaGaaiilaiaaysW7caWGybWaaSbaaKqbGeaacaWGUbaajuaG beaaaaa@4869@  be a random sample observed from zero-inflated truncated Poisson distribution truncated at 't MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacEcaca WG0bGaaGPaVdaa@39A8@  onwards, where 't MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacEcaca WG0bGaaGPaVdaa@39A8@  is the point in the support defined in the above probability mass function. Then the likelihood function is given by

L(θ,π; x _ )= i=1 n ( 1π+ π A(θ) ) 1 a i ( π θ x x!A(θ) ) a i θ,π>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeaca GGOaGaeqiUdeNaaiilaiabec8aWjaacUdadaadaaqaaiaadIhaaaGa aiykaiabg2da9maarahabaWaaeWaaeaacaaIXaGaeyOeI0IaeqiWda Naey4kaSYaaSaaaeaacqaHapaCaeaacaWGbbGaaiikaiabeI7aXjaa cMcaaaaacaGLOaGaayzkaaaajuaibaGaamyAaiabg2da9iaaigdaae aacaWGUbaajuaGcqGHpis1amaaCaaabeqcfasaaiaaigdacqGHsisl caWGHbqcfa4aaSbaaKqbGeaacaWGPbaabeaaaaqcfa4aaeWaaeaada Wcaaqaaiabec8aWjaaysW7cqaH4oqCdaahaaqabKqbGeaacaWG4baa aaqcfayaaiaadIhacaaMc8UaaiyiaiaadgeacaGGOaGaeqiUdeNaai ykaaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWGHbqcfa4aaSba aKqbGeaacaWGPbaabeaaaaqcfaOaaGzbVlaaywW7cqaH4oqCcaaMc8 UaaiilaiaaykW7cqaHapaCcqGH+aGpcaaIWaaaaa@75D4@  

The corresponding log likelihood function is given by

logL(θ,π; x _ )= n 0 log( 1π+ π A(θ) )+ i=1 n a i logπ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGacYgaca GGVbGaai4zaiaadYeacaGGOaGaeqiUdeNaaiilaiabec8aWjaacUda daadaaqaaiaadIhaaaGaaiykaiabg2da9iaad6gadaWgaaqcfasaai aaicdaaKqbagqaaiGacYgacaGGVbGaai4zamaabmaabaGaaGymaiab gkHiTiabec8aWjabgUcaRmaalaaabaGaeqiWdahabaGaamyqaiaacI cacqaH4oqCcaGGPaaaaaGaayjkaiaawMcaaiabgUcaRmaaqahabaGa amyyamaaBaaajuaibaGaamyAaaqcfayabaGaciiBaiaac+gacaGGNb GaeqiWdahajuaibaGaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaG cqGHris5aaaa@61DC@

+ i=1 n a i x i logθ i=1 n a i log x i ! i=1 n a i log A(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgUcaRm aaqahabaGaamyyamaaBaaajuaibaGaamyAaaqcfayabaGaamiEamaa BaaajuaibaGaamyAaaqcfayabaGaciiBaiaac+gacaGGNbGaeqiUde NaeyOeI0YaaabCaeaacaWGHbWaaSbaaKqbGeaacaWGPbaajuaGbeaa ciGGSbGaai4BaiaacEgacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbe aacaGGHaaajuaibaGaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaG cqGHris5aiabgkHiTmaaqahabaGaamyyamaaBaaajuaibaGaamyAaa qcfayabaGaciiBaiaac+gacaGGNbaajuaibaGaamyAaiabg2da9iaa igdaaeaacaWGUbaajuaGcqGHris5aiaadgeacaGGOaGaeqiUdeNaai ykaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyye Iuoaaaa@6910@  …(3.2)

To find MLEs of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj aaykW7aaa@39BA@  and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ , we differentiate the eq. (3.2) with respective π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXj aaykW7aaa@39BA@ , and then equating to zero we get

π ^ = (n n 0 )A( θ ^ ) n( A( θ ^ )1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0ZaaSaaaeaacaGGOaGaamOBaiabgkHiTiaad6gadaWg aaqcfasaaiaaicdaaKqbagqaaiaacMcacaWGbbGaaiikaiqbeI7aXz aajaGaaiykaaqaaiaad6gacaaMc8+aaeWaaeaacaWGbbGaaiikaiqb eI7aXzaajaGaaiykaiabgkHiTiaaykW7caaIXaaacaGLOaGaayzkaa aaaaaa@4E23@                                                      …(3.3)

and i=1 n a i x i θ ^ = π n 0 ( A ( θ ^ ) ) ( 1π+ π A( θ ^ ) ) ( A( θ ^ ) ) 2 + i=1 n a i A ( θ ^ ) A( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaabCaeaacaWGHbWaaSbaaKqbGeaacaWGPbaajuaGbeaacaWG4bWa aSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaaKqbakabggHiLdaabaGafqiUdeNbaKaaaaGaeyyp a0ZaaSaaaeaacqaHapaCcaaMe8UaamOBamaaBaaajuaibaGaaGimaa qcfayabaWaaeWaaeaaceWGbbGbauaacaGGOaGafqiUdeNbaKaacaGG PaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaIXaGaeyOeI0IaeqiWda Naey4kaSYaaSaaaeaacqaHapaCaeaacaWGbbGaaiikaiqbeI7aXzaa jaGaaiykaaaaaiaawIcacaGLPaaadaqadaqaaiaadgeacaGGOaGafq iUdeNbaKaacaGGPaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOm aaaaaaqcfaOaey4kaSYaaSaaaeaadaaeWbqaaiaadggadaWgaaqcfa saaiaadMgaaKqbagqaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGa amOBaaqcfaOaeyyeIuoaceWGbbGbauaacaGGOaGafqiUdeNbaKaaca GGPaaabaGaamyqaiaacIcacuaH4oqCgaqcaiaacMcaaaaaaa@753F@

= A ( θ ^ ) A( θ ^ ) ( (n n 0 )+ π ^ n 0 ( 1π+ π ^ A( θ ^ ) )A( θ ^ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGabmyqayaafaGaaiikaiqbeI7aXzaajaGaaiykaaqaaiaa dgeacaGGOaGafqiUdeNbaKaacaGGPaaaamaabmaabaGaaiikaiaad6 gacqGHsislcaWGUbWaaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPaGa ey4kaSYaaSaaaeaacuaHapaCgaqcaiaaysW7caWGUbWaaSbaaKqbGe aacaaIWaaajuaGbeaaaeaadaqadaqaaiaaigdacqGHsislcqaHapaC cqGHRaWkdaWcaaqaaiqbec8aWzaajaaabaGaamyqaiaacIcacuaH4o qCgaqcaiaacMcaaaaacaGLOaGaayzkaaGaamyqaiaacIcacuaH4oqC gaqcaiaacMcaaaaacaGLOaGaayzkaaaaaa@5CFE@

i=1 n a i x i θ = A (θ) A(θ) ( (n n 0 )(1π)A(θ)+nπ (1π)A(θ)+π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaabCaeaacaWGHbWaaSbaaKqbGeaacaWGPbaajuaGbeaacaWG4bWa aSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaaKqbakabggHiLdaabaGaeqiUdehaaiabg2da9maa laaabaGabmyqayaafaGaaiikaiabeI7aXjaacMcaaeaacaWGbbGaai ikaiabeI7aXjaacMcaaaWaaeWaaeaadaWcaaqaaiaacIcacaWGUbGa eyOeI0IaamOBamaaBaaajuaibaGaaGimaaqcfayabaGaaiykaiaacI cacaaIXaGaeyOeI0IaeqiWdaNaaiykaiaadgeacaGGOaGaeqiUdeNa aiykaiabgUcaRiaad6gacaaMc8UaeqiWdahabaGaaiikaiaaigdacq GHsislcqaHapaCcaGGPaGaamyqaiaacIcacqaH4oqCcaGGPaGaey4k aSIaeqiWdahaaaGaayjkaiaawMcaaaaa@6D4A@  …(3.4)

Substituting π ^ = (n n 0 )A( θ ^ ) n( A( θ ^ )1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0ZaaSaaaeaacaGGOaGaamOBaiabgkHiTiaad6gadaWg aaqcfasaaiaaicdaaKqbagqaaiaacMcacaWGbbGaaiikaiqbeI7aXz aajaGaaiykaaqaaiaad6gacaaMc8+aaeWaaeaacaWGbbGaaiikaiqb eI7aXzaajaGaaiykaiabgkHiTiaaykW7caaIXaaacaGLOaGaayzkaa aaaaaa@4E23@  in the above equation we have

i=1 n a i x i θ ^ = (n n 0 ) A ( θ ^ ) ( A( θ ^ )1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaabCaeaacaWGHbWaaSbaaKqbGeaacaWGPbaajuaGbeaacaWG4bWa aSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaaKqbakabggHiLdaabaGafqiUdeNbaKaaaaGaeyyp a0ZaaSaaaeaacaGGOaGaamOBaiabgkHiTiaad6gadaWgaaqcfasaai aaicdaaKqbagqaaiaacMcaceWGbbGbauaacaGGOaGafqiUdeNbaKaa caGGPaaabaWaaeWaaeaacaWGbbGaaiikaiqbeI7aXzaajaGaaiykai abgkHiTiaaigdaaiaawIcacaGLPaaaaaaaaa@5630@ ,

i=1 n a i x i ( A( θ ^ )1 )(n n 0 ) θ ^ A ( θ ^ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaqahaba GaamyyamaaBaaajuaibaGaamyAaaqcfayabaGaamiEamaaBaaajuai baGaamyAaaqcfayabaWaaeWaaeaacaWGbbGaaiikaiqbeI7aXzaaja GaaiykaiabgkHiTiaaigdaaiaawIcacaGLPaaacqGHsislcaGGOaGa amOBaiabgkHiTiaad6gadaWgaaqcfasaaiaaicdaaKqbagqaaiaacM cacaaMc8UafqiUdeNbaKaacaaMc8UabmyqayaafaGaaiikaiqbeI7a XzaajaGaaiykaiabg2da9iaaicdaaKqbGeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaaKqbakabggHiLdaaaa@5ACD@ , …(3.5)

which is non-linear equation in θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@ . Therefore, we use a numerical technique to solve it. Let

h( θ ^ )= i=1 n a i x i ( A( θ ^ )1 )(n n 0 ) θ ^ A ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIgaca GGOaGafqiUdeNbaKaacaGGPaGaeyypa0ZaaabCaeaacaWGHbWaaSba aKqbGeaacaWGPbaajuaGbeaacaWG4bWaaSbaaKqbGeaacaWGPbaaju aGbeaadaqadaqaaiaadgeacaGGOaGafqiUdeNbaKaacaGGPaGaeyOe I0IaaGymaaGaayjkaiaawMcaaiabgkHiTiaacIcacaWGUbGaeyOeI0 IaamOBamaaBaaajuaibaGaaGimaaqcfayabaGaaiykaiaaykW7cuaH 4oqCgaqcaiaaykW7ceWGbbGbauaacaGGOaGafqiUdeNbaKaacaGGPa aajuaibaGaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5 aaaa@5E1F@  and

h ( θ ^ )= i=1 n a i x i A ( θ ^ )(n n 0 )( θ ^ A ( θ ^ ) + A ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIgaga qbaiaacIcacuaH4oqCgaqcaiaacMcacqGH9aqpdaaeWbqaaiaadgga daWgaaqcfasaaiaadMgaaKqbagqaaiaadIhadaWgaaqcfasaaiaadM gaaKqbagqaaiqadgeagaqbaiaacIcacuaH4oqCgaqcaiaacMcacqGH sislcaGGOaGaamOBaiabgkHiTiaad6gadaWgaaqcfasaaiaaicdaaK qbagqaaiaacMcacaaMc8UaaGPaVpaabeaabaGafqiUdeNbaKaacaaM c8UabmyqayaafyaafaGaaiikaiqbeI7aXzaajaGaaiykaaGaayjkaa aajuaibaGaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5 aiabgUcaRiqadgeagaqbaiaacIcacuaH4oqCgaqcaiaacMcaaaa@6235@ .

Using Newton-Raphson iterative formula θ ^ i+1 = θ ^ i h( θ ^ ) h ( θ ^ ) ,i=0,1,2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaWaaSbaaKqbGeaacaWGPbGaey4kaSIaaGPaVlaaykW7caaIXaaa juaGbeaacqGH9aqpcuaH4oqCgaqcamaaBaaajuaibaGaamyAaaqcfa yabaGaeyOeI0YaaSaaaeaacaWGObGaaiikaiqbeI7aXzaajaGaaiyk aaqaaiqadIgagaqbaiaacIcacuaH4oqCgaqcaiaacMcaaaGaaiilai aaywW7caWGPbGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikda caGGSaGaaiOlaiaac6cacaGGUaaaaa@56FE@ with suitable initial value of θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXn aaBaaajuaibaGaaGimaaqcfayabaaaaa@39C6@  we get θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@ . Substituting this value of θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@  in eq. (3.3), we get the value of π ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaaaaa@3846@ .

In the following we find the elements of Fisher information matrix

Here we have

logL π = n 0 ( 1+ 1 A(θ) ) ( 1π+ π A(θ) ) + i=1 n a i π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaciiBaiaac+gacaGGNbGaamitaaqaaiabgkGi2kabec8a WbaacqGH9aqpdaWcaaqaaiaad6gadaWgaaqcfasaaiaaicdaaKqbag qaamaabmaabaGaeyOeI0IaaGymaiabgUcaRmaalaaabaGaaGymaaqa aiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaeaada qadaqaaiaaigdacqGHsislcqaHapaCcqGHRaWkdaWcaaqaaiabec8a WbqaaiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaa Gaey4kaSYaaSaaaeaadaaeWbqaaiaadggadaWgaaqcfasaaiaadMga aKqbagqaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfa OaeyyeIuoaaeaacqaHapaCaaaaaa@6260@ ,

2 logL π 2 = n 0 ( 1+ 1 A(θ) ) 2 ( 1π+ π A(θ) ) 2 i=1 n a i π 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGciGGSbGaai4Baiaa cEgacaWGmbaabaGaeyOaIyRaeqiWda3aaWbaaeqajuaibaGaaGOmaa aaaaqcfaOaeyypa0JaeyOeI0YaaSaaaeaacaWGUbWaaSbaaKqbGeaa caaIWaaajuaGbeaadaqadaqaaiabgkHiTiaaigdacqGHRaWkdaWcaa qaaiaaigdaaeaacaWGbbGaaiikaiabeI7aXjaacMcaaaaacaGLOaGa ayzkaaWaaWbaaeqajuaibaGaaGOmaaaaaKqbagaadaqadaqaaiaaig dacqGHsislcqaHapaCcqGHRaWkdaWcaaqaaiabec8aWbqaaiaadgea caGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaadaahaaqabKqbGe aacaaIYaaaaaaajuaGcqGHsisldaWcaaqaamaaqahabaGaamyyamaa BaaajuaibaGaamyAaaqcfayabaaajuaibaGaamyAaiabg2da9iaaig daaeaacaWGUbaajuaGcqGHris5aaqaaiabec8aWnaaCaaabeqcfasa aiaaikdaaaaaaaaa@6ACC@ ,

E( 2 logL π 2 )= E( n 0 ) ( 1+ 1 A(θ) ) 2 ( 1π+ π A(θ) ) 2 + E( i=1 n a i ) π 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalaaabaGaeyOaIy7aaWbaaeqajuaibaGaaGOm aaaajuaGciGGSbGaai4BaiaacEgacaWGmbaabaGaeyOaIyRaeqiWda 3aaWbaaeqajuaibaGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaGaeyyp a0ZaaSaaaeaacaWGfbGaaiikaiaad6gadaWgaaqcfasaaiaaicdaaK qbagqaaiaacMcadaqadaqaaiabgkHiTiaaigdacqGHRaWkdaWcaaqa aiaaigdaaeaacaWGbbGaaiikaiabeI7aXjaacMcaaaaacaGLOaGaay zkaaWaaWbaaeqajuaibaGaaGOmaaaaaKqbagaadaqadaqaaiaaigda cqGHsislcqaHapaCcqGHRaWkdaWcaaqaaiabec8aWbqaaiaadgeaca GGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaadaahaaqabKqbGeaa caaIYaaaaaaajuaGcqGHRaWkdaWcaaqaaiaadweadaqadaqaamaaqa habaGaamyyamaaBaaajuaibaGaamyAaaqcfayabaaajuaibaGaamyA aiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5aaGaayjkaiaawM caaaqaaiabec8aWnaaCaaabeqcfasaaiaaikdaaaaaaaaa@718A@ ,

= n ( 1+ 1 A(θ) ) 2 ( 1π+ π A(θ) ) + n( 1 1 A(θ) ) π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaamOBamaabmaabaGaeyOeI0IaaGymaiabgUcaRmaalaaa baGaaGymaaqaaiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcaca GLPaaadaahaaqabKqbGeaacaaIYaaaaaqcfayaamaabmaabaGaaGym aiabgkHiTiabec8aWjabgUcaRmaalaaabaGaeqiWdahabaGaamyqai aacIcacqaH4oqCcaGGPaaaaaGaayjkaiaawMcaaaaacqGHRaWkdaWc aaqaaiaad6gadaqadaqaaiaaigdacqGHsisldaWcaaqaaiaaigdaae aacaWGbbGaaiikaiabeI7aXjaacMcaaaaacaGLOaGaayzkaaaabaGa eqiWdahaaaaa@59B4@ ,

E( 2 logL π 2 )= n ( 1+ 1 A(θ) ) 2 ( 1π+ π A(θ) ) n( 1+ 1 A(θ) ) π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalaaabaGaeyOaIy7aaWbaaeqajuaibaGaaGOm aaaajuaGciGGSbGaai4BaiaacEgacaWGmbaabaGaeyOaIyRaeqiWda 3aaWbaaeqajuaibaGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaGaeyyp a0ZaaSaaaeaacaWGUbWaaeWaaeaacqGHsislcaaIXaGaey4kaSYaaS aaaeaacaaIXaaabaGaamyqaiaacIcacqaH4oqCcaGGPaaaaaGaayjk aiaawMcaamaaCaaabeqcfasaaiaaikdaaaaajuaGbaWaaeWaaeaaca aIXaGaeyOeI0IaeqiWdaNaey4kaSYaaSaaaeaacqaHapaCaeaacaWG bbGaaiikaiabeI7aXjaacMcaaaaacaGLOaGaayzkaaaaaiabgkHiTm aalaaabaGaamOBamaabmaabaGaeyOeI0IaaGymaiabgUcaRmaalaaa baGaaGymaaqaaiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcaca GLPaaaaeaacqaHapaCaaaaaa@694F@ ,                                   

I 11 = n( A(θ)1 ) π( A(θ)πA(θ)+π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaaigdacaaIXaaajuaGbeaacqGH9aqpdaWcaaqaaiaa d6gadaqadaqaaiaadgeacaGGOaGaeqiUdeNaaiykaiabgkHiTiaaig daaiaawIcacaGLPaaaaeaacqaHapaCdaqadaqaaiaadgeacaGGOaGa eqiUdeNaaiykaiabgkHiTiabec8aWjaadgeacaGGOaGaeqiUdeNaai ykaiabgUcaRiabec8aWbGaayjkaiaawMcaaaaacaaMc8oaaa@546D@ . …(3.6)

Now

logL π = n 0 ( 1+ 1 A(θ) ) ( 1π+ π A(θ) ) + i=1 n a i π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaciiBaiaac+gacaGGNbGaamitaaqaaiabgkGi2kabec8a WbaacqGH9aqpdaWcaaqaaiaad6gadaWgaaqcfasaaiaaicdaaKqbag qaamaabmaabaGaeyOeI0IaaGymaiabgUcaRmaalaaabaGaaGymaaqa aiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaeaada qadaqaaiaaigdacqGHsislcqaHapaCcqGHRaWkdaWcaaqaaiabec8a WbqaaiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaa Gaey4kaSYaaSaaaeaadaaeWbqaaiaadggadaWgaaqcfasaaiaadMga aeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIu oaaeaacqaHapaCaaaaaa@61A4@ ,

2 logL πθ = n 0 A (θ) ( A(θ) ) 2 ( ( 1π+ π A(θ) )π( 1+ 1 A(θ) ) ) ( 1π+ π A(θ) ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGciGGSbGaai4Baiaa cEgacaWGmbaabaGaeyOaIyRaeqiWdaNaaGPaVlabgkGi2kabeI7aXb aacqGH9aqpdaWcaaqaaiabgkHiTiaad6gadaWgaaqcfasaaiaaicda aKqbagqaamaalaaabaGabmyqayaafaGaaiikaiabeI7aXjaacMcaae aadaqadaqaaiaadgeacaGGOaGaeqiUdeNaaiykaaGaayjkaiaawMca amaaCaaabeqcfasaaiaaikdaaaaaaKqbaoaabmaabaWaaeWaaeaaca aIXaGaeyOeI0IaeqiWdaNaey4kaSYaaSaaaeaacqaHapaCaeaacaWG bbGaaiikaiabeI7aXjaacMcaaaaacaGLOaGaayzkaaGaeyOeI0Iaeq iWda3aaeWaaeaacqGHsislcaaIXaGaey4kaSYaaSaaaeaacaaIXaaa baGaamyqaiaacIcacqaH4oqCcaGGPaaaaaGaayjkaiaawMcaaaGaay jkaiaawMcaaaqaamaabmaabaGaaGymaiabgkHiTiabec8aWjabgUca RmaalaaabaGaeqiWdahabaGaamyqaiaacIcacqaH4oqCcaGGPaaaaa GaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@7940@ ,

E( 2 logL πθ )= E( n 0 )( A (θ) A (θ) 2 )( ( 1π+ π A(θ) )π( 1+ 1 A(θ) ) ) ( 1π+ π A(θ) ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalaaabaGaeyOaIy7aaWbaaeqajuaibaGaaGOm aaaajuaGciGGSbGaai4BaiaacEgacaWGmbaabaGaeyOaIyRaeqiWda NaaGPaVlabgkGi2kabeI7aXbaaaiaawIcacaGLPaaacqGH9aqpdaWc aaqaaiaadweacaGGOaGaamOBamaaBaaajuaibaGaaGimaaqcfayaba GaaiykamaabmaabaWaaSaaaeaaceWGbbGbauaacaGGOaGaeqiUdeNa aiykaaqaaiaadgeacaGGOaGaeqiUdeNaaiykamaaCaaabeqcfasaai aaikdaaaaaaaqcfaOaayjkaiaawMcaamaabmaabaWaaeWaaeaacaaI XaGaeyOeI0IaeqiWdaNaey4kaSYaaSaaaeaacqaHapaCaeaacaWGbb GaaiikaiabeI7aXjaacMcaaaaacaGLOaGaayzkaaGaeyOeI0IaeqiW da3aaeWaaeaacqGHsislcaaIXaGaey4kaSYaaSaaaeaacaaIXaaaba GaamyqaiaacIcacqaH4oqCcaGGPaaaaaGaayjkaiaawMcaaaGaayjk aiaawMcaaaqaamaabmaabaGaaGymaiabgkHiTiabec8aWjabgUcaRm aalaaabaGaeqiWdahabaGaamyqaiaacIcacqaH4oqCcaGGPaaaaaGa ayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@7DB6@ ,

E( 2 logL πθ )= I 12 = n A (θ) A(θ)( A(θ)πA(θ)+π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalaaabaGaeyOaIy7aaWbaaeqajuaibaGaaGOm aaaajuaGciGGSbGaai4BaiaacEgacaWGmbaabaGaeyOaIyRaeqiWda NaaGPaVlabgkGi2kabeI7aXbaaaiaawIcacaGLPaaacqGH9aqpcaWG jbWaaSbaaKqbGeaacaaIXaGaaGOmaaqcfayabaGaeyypa0ZaaSaaae aacaWGUbGaaGPaVlqadgeagaqbaiaacIcacqaH4oqCcaGGPaaabaGa amyqaiaacIcacqaH4oqCcaGGPaWaaeWaaeaacaWGbbGaaiikaiabeI 7aXjaacMcacqGHsislcqaHapaCcaWGbbGaaiikaiabeI7aXjaacMca cqGHRaWkcqaHapaCaiaawIcacaGLPaaaaaaaaa@6622@  …(3.7)

Further differentiating eq. (3.2) twice with respect to θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ , we get

2 logL θ 2 = n 0 π{ ( 1π+ π A(θ) )( A (θ) 2 A (θ)2 A (θ) 2 A(θ) f (θ) 4 ) A (θ) 2 A (θ) 2 ( π A (θ) 2 ) } ( 1π+ π A(θ) ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGciGGSbGaai4Baiaa cEgacaWGmbaabaGaeyOaIyRaeqiUde3aaWbaaeqajuaibaGaaGOmaa aaaaqcfaOaeyypa0JaeyOeI0YaaSaaaeaacaWGUbWaaSbaaKqbGeaa caaIWaaajuaGbeaacqaHapaCcaaMc8+aaiWaaeaadaqadaqaaiaaig dacqGHsislcqaHapaCcqGHRaWkdaWcaaqaaiabec8aWbqaaiaadgea caGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaadaqadaqaamaala aabaGaaGPaVlaadgeacaGGOaGaeqiUdeNaaiykamaaCaaabeqcfasa aiaaikdaaaqcfaOabmyqayaagaGaaiikaiabeI7aXjaacMcacqGHsi slcaaIYaGabmyqayaafaGaaiikaiabeI7aXjaacMcadaahaaqabKqb GeaacaaIYaaaaKqbakaadgeacaGGOaGaeqiUdeNaaiykaaqaaiaadA gacaGGOaGaeqiUdeNaaiykamaaCaaabeqcfasaaiaaisdaaaaaaaqc faOaayjkaiaawMcaaiabgkHiTmaalaaabaGabmyqayaafaGaaiikai abeI7aXjaacMcadaahaaqabKqbGeaacaaIYaaaaaqcfayaaiaadgea caGGOaGaeqiUdeNaaiykamaaCaaabeqcfasaaiaaikdaaaaaaKqbao aabmaabaGaeyOeI0YaaSaaaeaacqaHapaCcaaMc8oabaGaamyqaiaa cIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaGOmaaaaaaaajuaGca GLOaGaayzkaaaacaGL7bGaayzFaaaabaWaaeWaaeaacaaIXaGaeyOe I0IaeqiWdaNaey4kaSYaaSaaaeaacqaHapaCaeaacaWGbbGaaiikai abeI7aXjaacMcaaaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOm aaaaaaaaaa@97FE@

i=1 n a i x i θ 2 i=1 n a i ( A(θ) A (θ) A (θ) 2 ) A (θ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTm aalaaabaWaaabCaeaacaWGHbWaaSbaaKqbGeaacaWGPbaajuaGbeaa caWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGaey ypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaabaGaeqiUde3aaWba aeqajuaibaGaaGOmaaaaaaqcfaOaeyOeI0YaaSaaaeaadaaeWbqaai aadggadaWgaaqcfasaaiaadMgaaKqbagqaamaabmaabaGaamyqaiaa cIcacqaH4oqCcaGGPaGabmyqayaagaGaaiikaiabeI7aXjaacMcacq GHsislceWGbbGbauaacaGGOaGaeqiUdeNaaiykamaaCaaabeqcfasa aiaaikdaaaaajuaGcaGLOaGaayzkaaaajuaibaGaamyAaiabg2da9i aaigdaaeaacaWGUbaajuaGcqGHris5aaqaaiaadgeacaGGOaGaeqiU deNaaiykamaaCaaabeqcfasaaiaaikdaaaaaaaaa@6586@ .

Therefore,

E( 2 logL θ 2 )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalaaabaGaeyOaIy7aaWbaaeqajuaibaGaaGOm aaaajuaGciGGSbGaai4BaiaacEgacaWGmbaabaGaeyOaIyRaeqiUde 3aaWbaaeqajuaibaGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaGaeyyp a0daaa@4626@ ( nπ( ( A (θ) 2 A (θ)2 A (θ) 2 A(θ) A (θ) 4 )+ π A (θ) 2 A (θ) 4 ( 1π+ π A(θ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabeaaba GaamOBaiaaykW7cqaHapaCdaqadaqaamaabmaabaWaaSaaaeaacaWG bbGaaiikaiabeI7aXjaacMcadaahaaqabKqbGeaacaaIYaaaaKqbak qadgeagaGbaiaacIcacqaH4oqCcaGGPaGaeyOeI0IaaGOmaiqadgea gaqbaiaacIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaGOmaaaaju aGcaWGbbGaaiikaiabeI7aXjaacMcaaeaacaWGbbGaaiikaiabeI7a XjaacMcadaahaaqabKqbGeaacaaI0aaaaaaaaKqbakaawIcacaGLPa aacqGHRaWkdaWcaaqaaiabec8aWjaaysW7ceWGbbGbauaacaGGOaGa eqiUdeNaaiykamaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaamyqai aacIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaGinaaaajuaGdaqa daqaaiaaigdacqGHsislcqaHapaCcqGHRaWkdaWcaaqaaiabec8aWb qaaiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaaaa caGLOaGaayzkaaaacaGLOaaaaaa@72F1@

+ nπ A (θ) θA(θ) + nπ( 1 1 A(θ) )( A(θ) A (θ) A (θ) 2 ) A (θ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabiaaba Gaey4kaSYaaSaaaeaacaWGUbGaaGPaVlabec8aWjaaysW7ceWGbbGb auaacaGGOaGaeqiUdeNaaiykaaqaaiabeI7aXjaaysW7caWGbbGaai ikaiabeI7aXjaacMcaaaGaey4kaSYaaSaaaeaacaWGUbGaaGPaVlab ec8aWnaabmaabaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaaiaadg eacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaadaqadaqaaiaa dgeacaGGOaGaeqiUdeNaaiykaiqadgeagaGbaiaacIcacqaH4oqCca GGPaGaeyOeI0IabmyqayaafaGaaiikaiabeI7aXjaacMcadaahaaqa bKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaaqaaiaadgeacaGGOa GaeqiUdeNaaiykamaaCaaabeqcfasaaiaaikdaaaaaaaqcfaOaayzk aaaaaa@6B09@ .

Hence,

                I 22 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIYaGaaGOmaaqabaGccqGH9aqpaaa@3979@ ( nπ( ( A (θ) 2 A (θ)2 A (θ) 2 A(θ) A (θ) 4 )+ π A (θ) 2 A (θ) 4 ( 1π+ π A(θ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabeaaba GaamOBaiaaykW7cqaHapaCdaqadaqaamaabmaabaWaaSaaaeaacaWG bbGaaiikaiabeI7aXjaacMcadaahaaqabKqbGeaacaaIYaaaaKqbak qadgeagaGbaiaacIcacqaH4oqCcaGGPaGaeyOeI0IaaGOmaiqadgea gaqbaiaacIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaGOmaaaaju aGcaWGbbGaaiikaiabeI7aXjaacMcaaeaacaWGbbGaaiikaiabeI7a XjaacMcadaahaaqabKqbGeaacaaI0aaaaaaaaKqbakaawIcacaGLPa aacqGHRaWkdaWcaaqaaiabec8aWjaaysW7ceWGbbGbauaacaGGOaGa eqiUdeNaaiykamaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaamyqai aacIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaGinaaaajuaGdaqa daqaaiaaigdacqGHsislcqaHapaCcqGHRaWkdaWcaaqaaiabec8aWb qaaiaadgeacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaaaa caGLOaGaayzkaaaacaGLOaaaaaa@72F1@

+ nπ A (θ) θA(θ) + nπ( 1 1 A(θ) )( A(θ) A (θ) A (θ) 2 ) A (θ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabiaaba Gaey4kaSYaaSaaaeaacaWGUbGaaGPaVlabec8aWjaaysW7ceWGbbGb auaacaGGOaGaeqiUdeNaaiykaaqaaiabeI7aXjaaysW7caWGbbGaai ikaiabeI7aXjaacMcaaaGaey4kaSYaaSaaaeaacaWGUbGaaGPaVlab ec8aWnaabmaabaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaaiaadg eacaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaadaqadaqaaiaa dgeacaGGOaGaeqiUdeNaaiykaiqadgeagaGbaiaacIcacqaH4oqCca GGPaGaeyOeI0IabmyqayaafaGaaiikaiabeI7aXjaacMcadaahaaqa bKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaaqaaiaadgeacaGGOa GaeqiUdeNaaiykamaaCaaabeqcfasaaiaaikdaaaaaaaqcfaOaayzk aaaaaa@6B09@ .

The asymptotic variance of π ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aaaaa@37C4@  and θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BD@  are

A V π ^ (π,θ)= I 11 = ( I 11 I 12 I 22 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaHapaCgaqcaaqcfayabaGaaiikaiabec8a WjaacYcacqaH4oqCcaGGPaGaeyypa0JaamysamaaCaaabeqcfasaai aaigdacaaIXaaaaKqbakabg2da9maabmaabaGaamysamaaBaaajuai baGaaGymaiaaigdaaKqbagqaaiabgkHiTmaalaaabaGaamysamaaBa aajuaibaGaaGymaiaaikdaaKqbagqaaaqaaiaadMeadaWgaaqcfasa aiaaikdacaaIYaaajuaGbeaaaaaacaGLOaGaayzkaaWaaWbaaeqaju aibaGaeyOeI0IaaGymaaaaaaa@5352@ .

A V θ ^ (π,θ)= I 22 = ( I 22 I 12 I 11 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaqcaaqcfayabaGaaiikaiabec8a WjaacYcacqaH4oqCcaGGPaGaeyypa0JaamysamaaCaaabeqcfasaai aaikdacaaIYaaaaKqbakabg2da9maabmaabaGaamysamaaBaaajuai baGaaGOmaiaaikdaaKqbagqaaiabgkHiTmaalaaabaGaamysamaaBa aajuaibaGaaGymaiaaikdaaKqbagqaaaqaaiaadMeadaWgaaqcfasa aiaaigdacaaIXaaajuaGbeaaaaaacaGLOaGaayzkaaWaaWbaaeqaju aibaGaeyOeI0IaaGymaaaaaaa@534D@  .                                                … (3.8)

  1. Conditional likelihood function approach

The conditional likelihood function is given by

L (θ; x _ )= i=1 n ( θ x i x i !(A(θ)1) ) a i , θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada ahaaqabKqbGeaacqGHxiIkaaqcfaOaaiikaiabeI7aXjaaykW7caGG 7aWaaWaaaeaacaWG4baaaiaaykW7caGGPaGaeyypa0ZaaebCaeaada qadaqaamaalaaabaGaaGPaVlabeI7aXnaaCaaabeqaaiaadIhadaWg aaqaaiaadMgaaeqaaaaaaeaacaWG4bWaaSbaaKqbGeaacaWGPbaaju aGbeaacaGGHaGaaGPaVlaacIcacaWGbbGaaiikaiabeI7aXjaacMca cqGHsislcaaIXaGaaiykaaaaaiaawIcacaGLPaaadaahaaqabKqbGe aacaWGHbqcfa4aaSbaaKqbGeaacaWGPbaabeaaaaqcfaOaaiilaaqc fasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaey4dIunaca aMf8UaaGzbVlabeI7aXjaaykW7caaMc8UaeyOpa4JaaGimaaaa@6944@                   …(3.9)

The corresponding log likelihood function is given by

logL*(θ; x _ )= i=1 n n 0 x i log(θ)(n n 0 )log(A(θ)1) i=1 n n 0 log x i ! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakGacYgaca GGVbGaai4zaiaadYeacaaMc8UaaiOkaiaacIcacqaH4oqCcaaMc8Ua ai4oamaamaaabaGaamiEaaaacaGGPaGaeyypa0ZaaabCaeaacaWG4b WaaSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGaeyypa0Ja aGymaaqaaiaad6gacqGHsislcaWGUbqcfa4aaSbaaKqbGeaacaaIWa aabeaaaKqbakabggHiLdGaciiBaiaac+gacaGGNbGaaiikaiabeI7a XjaacMcacqGHsislcaGGOaGaamOBaiabgkHiTiaad6gadaWgaaqcfa saaiaaicdaaKqbagqaaiaacMcaciGGSbGaai4BaiaacEgacaGGOaGa amyqaiaacIcacqaH4oqCcaGGPaGaeyOeI0IaaGymaiaacMcacqGHsi sldaaeWbqaaiGacYgacaGGVbGaai4zaiaadIhadaWgaaqcfasaaiaa dMgaaKqbagqaaiaacgcaaKqbGeaacaWGPbGaeyypa0JaaGymaaqaai aad6gacqGHsislcaWGUbqcfa4aaSbaaKqbGeaacaaIWaaabeaaaKqb akabggHiLdaaaa@77E4@  …(3.10)

The corresponding mle θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaiaaaaa@383E@  is the solution to an equation

x ¯ = θ ˜ A( θ ˜ ) A( θ ˜ )1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qeaiabg2da9maalaaabaGafqiUdeNbaGaacaWGbbGaaiikaiqbeI7a XzaaiaGaaiykaaqaaiaadgeacaGGOaGafqiUdeNbaGaacaGGPaGaey OeI0IaaGymaaaaaaa@43D9@  …(3.11)

Now consider,

2 log L * θ 2 = i=1 n n 0 x i θ 2 i=1 n n 0 ( A(θ)1 ) A (θ) A (θ) 2 ( A(θ)1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGciGGSbGaai4Baiaa cEgacaWGmbWaaWbaaeqajuaibaGaaiOkaaaaaKqbagaacqGHciITcq aH4oqCdaahaaqabKqbGeaacaaIYaaaaaaajuaGcqGH9aqpcqGHsisl daaeWbqaamaalaaabaGaamiEamaaBaaajuaibaGaamyAaaqcfayaba aabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaaaabaGaamyAaiab g2da9iaaigdaaeaacaWGUbGaeyOeI0IaamOBaKqbaoaaBaaajuaiba GaaGimaaqabaaajuaGcqGHris5aiabgkHiTmaaqahabaWaaSaaaeaa daqadaqaaiaadgeacaGGOaGaeqiUdeNaaiykaiabgkHiTiaaigdaai aawIcacaGLPaaaceWGbbGbayaacaGGOaGaeqiUdeNaaiykaiabgkHi TiqadgeagaqbaiaacIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaG OmaaaaaKqbagaadaqadaqaaiaadgeacaGGOaGaeqiUdeNaaiykaiab gkHiTiaaigdaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaa aaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gacqGHsislcaWGUbqc fa4aaSbaaKqbGeaacaaIWaaabeaaaKqbakabggHiLdaaaa@78FA@

E( 2 log L * θ 2 )=E( i=1 n n 0 x i θ 2 )+E( i=1 n n 0 ( A(θ)1 ) A (θ) A (θ) 2 ( A(θ)1 ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalaaabaGaeyOaIy7aaWbaaeqajuaibaGaaGOm aaaajuaGciGGSbGaai4BaiaacEgacaWGmbWaaWbaaeqabaGaaiOkaa aaaeaacqGHciITcqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaaaaKqb akaawIcacaGLPaaacqGH9aqpcaWGfbWaaeWaaeaadaaeWbqaamaala aabaGaamiEamaaBaaajuaibaGaamyAaaqcfayabaaabaGaeqiUde3a aWbaaeqajuaibaGaaGOmaaaaaaaabaGaamyAaiabg2da9iaaigdaaK qbagaajuaicaWGUbGaeyOeI0IaamOBaKqbaoaaBaaajuaibaGaaGim aaqcfayabaaacqGHris5aaGaayjkaiaawMcaaiabgUcaRiaadweada qadaqaamaaqahabaWaaSaaaeaadaqadaqaaiaadgeacaGGOaGaeqiU deNaaiykaiabgkHiTiaaigdaaiaawIcacaGLPaaaceWGbbGbayaaca GGOaGaeqiUdeNaaiykaiabgkHiTiqadgeagaqbaiaacIcacqaH4oqC caGGPaWaaWbaaeqajuaibaGaaGOmaaaaaKqbagaadaqadaqaaiaadg eacaGGOaGaeqiUdeNaaiykaiabgkHiTiaaigdaaiaawIcacaGLPaaa daahaaqabKqbGeaacaaIYaaaaaaaaeaacaWGPbGaeyypa0JaaGymaa qcfayaaKqbGiaad6gacqGHsislcaWGUbqcfa4aaSbaaKqbGeaacaaI WaaajuaGbeaaaiabggHiLdaacaGLOaGaayzkaaaaaa@80A4@

= (n n 0 )θ A (θ) θ 2 (A(θ)1) + (n n 0 ){ (A(θ)1) A (θ) A (θ) 2 } ( A(θ)1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaaiikaiaad6gacqGHsislcaWGUbWaaSbaaKqbGeaacaaI WaaajuaGbeaacaGGPaGaeqiUdeNaaGPaVlqadgeagaqbaiaacIcacq aH4oqCcaGGPaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaG caGGOaGaamyqaiaacIcacqaH4oqCcaGGPaGaeyOeI0IaaGymaiaacM caaaGaey4kaSYaaSaaaeaacaGGOaGaamOBaiabgkHiTiaad6gadaWg aaqcfasaaiaaicdaaKqbagqaaiaacMcadaGadaqaaiaacIcacaWGbb GaaiikaiabeI7aXjaacMcacqGHsislcaaIXaGaaiykaiqadgeagaGb aiaacIcacqaH4oqCcaGGPaGaeyOeI0IabmyqayaafaGaaiikaiabeI 7aXjaacMcadaahaaqabKqbGeaacaaIYaaaaaqcfaOaay5Eaiaaw2ha aaqaamaabmaabaGaamyqaiaacIcacqaH4oqCcaGGPaGaeyOeI0IaaG ymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@70B2@

= (n n 0 ) (A(θ)1) ( A (θ) θ + A (θ)(A(θ)1) A (θ) 2 (A(θ)1) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aalaaabaGaaiikaiaad6gacqGHsislcaWGUbWaaSbaaKqbGeaacaaI WaaajuaGbeaacaGGPaaabaGaaiikaiaadgeacaGGOaGaeqiUdeNaai ykaiabgkHiTiaaigdacaGGPaaaamaabmaabaWaaSaaaeaacaaMc8Ua bmyqayaafaGaaiikaiabeI7aXjaacMcaaeaacqaH4oqCaaGaey4kaS YaaSaaaeaaceWGbbGbayaacaGGOaGaeqiUdeNaaiykaiaacIcacaWG bbGaaiikaiabeI7aXjaacMcacqGHsislcaaIXaGaaiykaiabgkHiTi qadgeagaqbaiaacIcacqaH4oqCcaGGPaWaaWbaaeqajuaibaGaaGOm aaaaaKqbagaacaGGOaGaamyqaiaacIcacqaH4oqCcaGGPaGaeyOeI0 IaaGymaiaacMcaaaaacaGLOaGaayzkaaaaaa@65CB@ . … (3.12)

Therefore, asymptotic variance of θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aaaaa@37BC@  is different than the asymptotic variance of estimate of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  based on the standard likelihood approach. The same is given by

A V θ ˜ (θ)= ( (n n 0 ) ( A(θ)1 ) ( A (θ) θ + (A(θ)1)) A (θ) ( A (θ)) 2 ( A(θ)1 ) ) ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiabeI7a XjaacMcacqGH9aqpdaqadaqaamaalaaabaGaaGPaVlaacIcacaWGUb GaeyOeI0IaamOBamaaBaaajuaibaGaaGimaaqcfayabaGaaiykaaqa aiaaykW7caaMc8+aaeWaaeaacaWGbbGaaiikaiabeI7aXjaacMcacq GHsislcaaIXaaacaGLOaGaayzkaaaaamaabmaabaWaaSaaaeaaceWG bbGbauaacaGGOaGaeqiUdeNaaiykaaqaaiabeI7aXbaacqGHRaWkda WcaaqaaiaacIcacaWGbbGaaiikaiabeI7aXjaacMcacqGHsislcaaI XaGaaiykaiaacMcaceWGbbGbayaacaGGOaGaeqiUdeNaaiykaiabgk HiTiaacIcaceWGbbGbauaacaGGOaGaeqiUdeNaaiykaiaacMcadaah aaqabKqbGeaacaaIYaaaaaqcfayaamaabmaabaGaamyqaiaacIcacq aH4oqCcaGGPaGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaaaiaawIca caGLPaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacqGHsislcaaIXa aaaaaa@761A@  … (3.13)

  1. Moment estimator of ZITP distribution

 Mean = E(X)= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweaca GGOaGaamiwaiaacMcacqGH9aqpaaa@3A7F@ πθ A (θ) A(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeqiWdaNaaGPaVlabeI7aXjaaysW7ceWGbbGbauaacaGGOaGaeqiU deNaaGjbVlaacMcaaeaacaWGbbGaaiikaiabeI7aXjaacMcaaaGaaG jbVdaa@47E4@                                  …(3.14)

E( X 2 )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweaca GGOaGaamiwamaaCaaabeqcfasaaiaaikdaaaqcfaOaaiykaiabg2da 9aaa@3C19@ θπ A(θ) ( θ A (θ)+ A (θ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeqiUdeNaaGjbVlabec8aWbqaaiaadgeacaGGOaGaeqiUdeNaaiyk aaaadaqadaqaaiabeI7aXjaaysW7ceWGbbGbayaacaGGOaGaeqiUde NaaGjbVlaacMcacaaMe8Uaey4kaSIabmyqayaafaGaaiikaiabeI7a XjaaysW7caGGPaaacaGLOaGaayzkaaaaaa@5176@

Var(X)= θπ A(θ) ( θ A (θ)+ A (θ) πθ A (θ) 2 A(θ) )= σ 2 (π,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfaca WGHbGaamOCaiaacIcacaWGybGaaiykaiabg2da9maalaaabaGaeqiU deNaaGjbVlabec8aWbqaaiaadgeacaGGOaGaeqiUdeNaaiykaaaada qadaqaaiabeI7aXjaaysW7ceWGbbGbayaacaGGOaGaeqiUdeNaaGjb VlaacMcacaaMe8Uaey4kaSIabmyqayaafaGaaiikaiabeI7aXjaays W7caGGPaGaeyOeI0YaaSaaaeaacaaMe8UaeqiWdaNaaGjbVlabeI7a XjaaykW7ceWGbbGbauaacaGGOaGaeqiUdeNaaiykamaaCaaabeqcfa saaiaaikdaaaaajuaGbaGaamyqaiaacIcacqaH4oqCcaGGPaaaaaGa ayjkaiaawMcaaiabg2da9iabeo8aZnaaCaaabeqcfasaaiaaikdaaa qcfaOaaiikaiabec8aWjaacYcacqaH4oqCcaGGPaaaaa@73AE@  say       …(3.15)

x ¯ = πθ A (θ) A(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qeaiabg2da9maalaaabaGaeqiWdaNaaGPaVlabeI7aXjaaysW7ceWG bbGbauaacaGGOaGaeqiUdeNaaGjbVlaacMcaaeaacaWGbbGaaiikai abeI7aXjaacMcaaaGaaGjbVdaa@49FF@  …(3.16)

i=1 n x i 2 n = θπ A(θ) ( θ A (θ)+ A (θ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaabCaeaacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaadaahaaqa bKqbGeaacaaIYaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaa qcfaOaeyyeIuoaaeaacaWGUbaaaiabg2da9maalaaabaGaeqiUdeNa aGjbVlabec8aWbqaaiaadgeacaGGOaGaeqiUdeNaaiykaaaadaqada qaaiabeI7aXjaaysW7ceWGbbGbayaacaGGOaGaeqiUdeNaaGjbVlaa cMcacaaMe8Uaey4kaSIabmyqayaafaGaaiikaiabeI7aXjaaysW7ca GGPaaacaGLOaGaayzkaaaaaa@5DAD@  …(3.17)

Solving eq. (3.16) and eq. (3.17), we get moment estimators of π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ .

Tests for the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  of ZITP distribution

Suppose we want to test H 0 :θ= θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaiaacQdacqaH4oqCcqGH9aqpcqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaaa@3FA4@  vs H 1 :θ θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaigdaaKqbagqaaiaacQdacqaH4oqCcqGHGjsUcqaH 4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaaa@4066@ , (assuming π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  is unknown) [4]

  1. Test based on θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@

Z 4 = θ ^ θ 0 A V θ ^ ( π ^ 0 , θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaisdaaKqbagqaaiabg2da9maalaaabaGafqiUdeNb aKaacqGHsislcqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaqaam aakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaajaaajuaG beaacaGGOaGafqiWdaNbaKaadaWgaaqcfasaaiaaicdaaKqbagqaai aacYcacqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaqabaGaaiyk aaaaaaa@4D17@                                   …(3.18)

where A V θ ^ ( π ^ 0 , θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaqcaaqcfayabaGaaiikaiqbec8a WzaajaWaaSbaaKqbGeaacaaIWaaajuaGbeaacaGGSaGaeqiUde3aaS baaKqbGeaacaaIWaaajuaGbeaacaGGPaaaaa@4377@ is defined in eq. (3.8).The test ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@  rejects H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@ , if | Z 4 |> z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GaamOwamaaBaaajuaibaGaaGinaaqcfayabaaacaGLhWUaayjcSdGa eyOpa4JaamOEamaaBaaajuaibaGaaGymaiabgkHiTiabeg7aHjaac+ cacaaIYaaajuaGbeaaaaa@43AF@ .

  1. Test based on θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@

 The test statistic here is Z 5 = θ ˜ θ 0 A V θ ˜ ( θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaiwdaaKqbagqaaiabg2da9maalaaabaGafqiUdeNb aGaacqGHsislcqaH4oqCdaWgaaqcfasaaiaaicdaaKqbagqaaaqaam aakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaaiaaajuaG beaacaGGOaGaeqiUde3aaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPa GaaGjbVdqabaaaaaaa@4A8F@ , …(3.19)

Where, A V θ ˜ ( θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiabeI7a XnaaBaaajuaibaGaaGimaaqcfayabaGaaiykaaaa@3F62@  is as defined in eq. (3.13). The test ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@  rejects H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcfasaaiaaicdaaKqbagqaaaaa@38DD@  if | Z 5 |> z 1α/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaemaaba GaamOwamaaBaaajuaibaGaaGynaaqcfayabaaacaGLhWUaayjcSdGa eyOpa4JaamOEamaaBaaajuaibaGaaGymaiabgkHiTiabeg7aHjaac+ cacaaIYaaajuaGbeaaaaa@43B0@ .

  1. Test based on sample mean

The test statistic

Z 6 = n ( X ¯ π ^ 0 θ 0 ) π ^ 0 2 A V X ¯ ( π ^ 0 , θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQfada WgaaqcfasaaiaaiAdaaKqbagqaaiabg2da9maalaaabaWaaOaaaeaa caWGUbGaaGjbVdqabaWaaeWaaeaadaWcaaqaaiqadIfagaqeaaqaai qbec8aWzaajaWaaSbaaKqbGeaacaaIWaaajuaGbeaaaaGaeyOeI0Ia eqiUde3aaSbaaKqbGeaacaaIWaaajuaGbeaaaiaawIcacaGLPaaaae aadaGcaaqaaiqbec8aWzaajaWaaSbaaKqbGeaacaaIWaaajuaGbeaa daahaaqabKqbGeaacqGHsislcaaIYaaaaKqbakaadgeacaWGwbWaaS baaKqbGeaaceWGybGbaebaaKqbagqaaiaacIcacuaHapaCgaqcamaa BaaajuaibaGaaGimaaqcfayabaGaaiilaiabeI7aXnaaBaaajuaiba GaaGimaaqcfayabaGaaiykaaqabaaaaaaa@58EF@  , …(3.20)

where π ^ 0 = X ¯ ( A( θ 0 ) θ 0 A ( θ 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGH9aqpceWGybGbaeba daqadaqaamaalaaabaGaamyqaiaacIcacqaH4oqCdaWgaaqcfasaai aaicdaaKqbagqaaiaacMcaaeaacqaH4oqCdaWgaaqcfasaaiaaicda aKqbagqaaiqadgeagaqbaiaacIcacqaH4oqCdaWgaaqcfasaaiaaic daaKqbagqaaiaacMcaaaaacaGLOaGaayzkaaaaaa@4BA2@

Power of the test is given by

β ψ 6 ( π ^ ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGaeqiYdKxcfa4aaSbaaKqbGeaacaaI2aaabeaaaKqb agqaaiaacIcacuaHapaCgaqcaiaacYcacqaH4oqCcaGGPaaaaa@41EE@   = k=0 n ( 1Φ( B ^ k )+Φ( A ^ k ) )P( n 0 =k) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabg2da9m aaqahabaWaaeWaaeaacaaIXaGaeyOeI0IaeuOPdyKaaiikaiqadkea gaqcamaaBaaajuaibaGaam4AaaqcfayabaGaaiykaiabgUcaRiabfA 6agjaacIcaceWGbbGbaKaadaWgaaqcfasaaiaadUgaaKqbagqaaiaa cMcaaiaawIcacaGLPaaacaWGqbGaaiikaiaad6gadaWgaaqcfasaai aaicdaaKqbagqaaiabg2da9iaadUgacaGGPaaajuaibaGaam4Aaiab g2da9iaaicdaaeaacaWGUbaajuaGcqGHris5aaaa@53B6@

where , B ^ k = π ^ 0 ( θ 0 + z 1α/2 π ^ 0 2 A V X ¯ ( π ^ 0 , θ 0 ) )πθ A V X ¯ (π, θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadkeaga qcamaaBaaajuaibaGaam4AaaqcfayabaGaeyypa0ZaaSaaaeaacuaH apaCgaqcamaaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacqaH4o qCdaWgaaqcfasaaiaaicdaaKqbagqaaiabgUcaRiaadQhadaWgaaqc fasaaiaaigdacqGHsislcqaHXoqycaGGVaGaaGOmaaqcfayabaWaaO aaaeaacuaHapaCgaqcamaaBaaajuaibaGaaGimaaqcfayabaWaaWba aeqajuaibaGaeyOeI0IaaGOmaaaajuaGcaWGbbGaamOvamaaBaaaju aibaGabmiwayaaraaajuaGbeaacaGGOaGafqiWdaNbaKaadaWgaaqc fasaaiaaicdaaKqbagqaaiaacYcacqaH4oqCdaWgaaqaaiaaicdaae qaaiaacMcaaeqaaaGaayjkaiaawMcaaiabgkHiTiabec8aWjaaykW7 cqaH4oqCaeaadaGcaaqaaiaadgeacaWGwbWaaSbaaKqbGeaaceWGyb GbaebaaKqbagqaaiaacIcacqaHapaCcaGGSaGaeqiUde3aaSbaaKqb GeaacaaIWaaajuaGbeaacaGGPaaabeaaaaaaaa@6BCE@ ,

A ^ k = π ^ 0 ( θ 0 z 1α/2 π ^ 0 2 A V X ¯ ( π ^ , θ 0 ) )πθ A V X ¯ (π, θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadgeaga qcamaaBaaajuaibaGaam4AaaqcfayabaGaeyypa0ZaaSaaaeaacuaH apaCgaqcamaaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacqaH4o qCdaWgaaqcfasaaiaaicdaaKqbagqaaiabgkHiTiaadQhadaWgaaqc fasaaiaaigdacqGHsislcqaHXoqycaGGVaGaaGOmaaqcfayabaWaaO aaaeaacuaHapaCgaqcamaaBaaajuaibaGaaGimaaqcfayabaWaaWba aeqajuaibaGaeyOeI0IaaGOmaaaajuaGcaWGbbGaamOvamaaBaaaju aibaGabmiwayaaraaajuaGbeaacaGGOaGafqiWdaNbaKaacaGGSaGa eqiUde3aaSbaaKqbGeaacaaIWaaajuaGbeaacaGGPaaabeaaaiaawI cacaGLPaaacqGHsislcqaHapaCcaaMc8UaeqiUdehabaWaaOaaaeaa caWGbbGaamOvamaaBaaajuaibaGabmiwayaaraaajuaGbeaacaGGOa GaeqiWdaNaaiilaiabeI7aXnaaBaaajuaibaGaaGimaaqcfayabaGa aiykaaqabaaaaaaa@6AFD@  and

P( n 0 =k)=( k n ) P 0 k (1 P 0 ) nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfaca GGOaGaamOBamaaBaaajuaibaGaaGimaaqcfayabaGaeyypa0Jaam4A aiaacMcacqGH9aqpdaqadaqaamaaDeaabaGaam4Aaaqaaiaad6gaaa aacaGLOaGaayzkaaGaamiuamaaBaaajuaibaGaaGimaaqcfayabaWa aWbaaeqajuaibaGaam4AaaaajuaGcaGGOaGaaGymaiabgkHiTiaadc fadaWgaaqcfasaaiaaicdaaKqbagqaaiaacMcadaahaaqabKqbGeaa caWGUbGaeyOeI0Iaam4AaaaajuaGcaaMc8UaaGPaVdaa@5227@ , with P 0 =1π+ π A(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadcfada WgaaqcfasaaiaaicdaaKqbagqaaiabg2da9iaaigdacqGHsislcqaH apaCcqGHRaWkdaWcaaqaaiabec8aWbqaaiaadgeacaGGOaGaeqiUde Naaiykaaaaaaa@43D4@

Asymptotic confidence interval for the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@

Asymptotic confidence interval for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  based on the test ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@  is given by

( θ ^ z 1α/2 A V θ ^ ( π ^ , θ ^ ) , θ ^ + z 1α/2 A V θ ^ ( π ^ , θ ^ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GafqiUdeNbaKaacqGHsislcaWG6bWaaSbaaKqbGeaacaaIXaGaeyOe I0IaeqySdeMaai4laiaaikdaaKqbagqaamaakaaabaGaamyqaiaadA fadaWgaaqcfasaaiqbeI7aXzaajaaajuaGbeaacaGGOaGafqiWdaNb aKaacaGGSaGafqiUdeNbaKaacaGGPaaabeaacaGGSaGaaGPaVlaayw W7cuaH4oqCgaqcaiabgUcaRiaadQhadaWgaaqcfasaaiaaigdacqGH sislcqaHXoqycaGGVaGaaGOmaaqcfayabaWaaOaaaeaacaWGbbGaam OvamaaBaaajuaibaGafqiUdeNbaKaaaKqbagqaaiaacIcacuaHapaC gaqcaiaacYcacuaH4oqCgaqcaiaacMcaaeqaaaGaayjkaiaawMcaaa aa@622A@  …(3.21)

where, A V θ ^ ( π ^ , θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaqcaaqcfayabaGaaiikaiqbec8a WzaajaGaaiilaiqbeI7aXzaajaGaaiykaaaa@4059@  is an estimate of asymptotic variance of θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaaaaa@383F@  and asymptotic confidence interval for q based on the test ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@  is given by

( θ ˜ z 1α/2 A V θ ˜ ( θ ˜ ) , θ ˜ + z 1α/2 A V θ ˜ ( θ ˜ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GafqiUdeNbaGaacqGHsislcaWG6bWaaSbaaKqbGeaacaaIXaGaeyOe I0IaeqySdeMaai4laiaaikdaaKqbagqaamaakaaabaGaamyqaiaadA fadaWgaaqcfasaaiqbeI7aXzaaiaaajuaGbeaacaGGOaGafqiUdeNb aGaacaGGPaaabeaacaGGSaGaaGPaVlaaywW7cuaH4oqCgaacaiabgU caRiaadQhadaWgaaqcfasaaiaaigdacqGHsislcqaHXoqycaGGVaGa aGOmaaqcfayabaWaaOaaaeaacaWGbbGaamOvamaaBaaajuaibaGafq iUdeNbaGaaaKqbagqaaiaacIcacuaH4oqCgaacaiaacMcaaeqaaaGa ayjkaiaawMcaaaaa@5D2A@  …(3.22)

where A V θ ˜ ( θ ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaacaaqcfayabaGaaiikaiqbeI7a XzaaiaGaaiykaaaa@3DDA@  is an estimate of the asymptotic variance of θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aaiaaaaa@383E@  as given in the eq. (3.13) .

Asymptotic confidence interval for θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@  based on the test ψ 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGOnaaqcfayabaaaaa@39E4@  is given by

( X ¯ π ^ z 1α/2 A V θ ¯ ( π ^ , θ ¯ ) , X ¯ π ^ + z 1α/2 A V θ ¯ ( π ^ , θ ¯ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba WaaSaaaeaaceWGybGbaebaaeaacuaHapaCgaqcaaaacqGHsislcaWG 6bWaaSbaaKqbGeaacaaIXaGaeyOeI0IaeqySdeMaai4laiaaikdaaK qbagqaamaakaaabaGaamyqaiaadAfadaWgaaqcfasaaiqbeI7aXzaa raaajuaGbeaacaGGOaGafqiWdaNbaKaacaGGSaGafqiUdeNbaebaca GGPaaabeaacaGGSaGaaGPaVlaaywW7daWcaaqaaiqadIfagaqeaaqa aiqbec8aWzaajaaaaiabgUcaRiaadQhadaWgaaqcfasaaiaaigdacq GHsislcqaHXoqycaGGVaGaaGOmaaqcfayabaWaaOaaaeaacaWGbbGa amOvamaaBaaajuaibaGafqiUdeNbaebaaKqbagqaaiaacIcacuaHap aCgaqcaiaacYcacuaH4oqCgaqeaiaacMcaaeqaaaGaayjkaiaawMca aaaa@6462@ , …(3.23)

where A V θ ¯ ( π ^ , θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGwbWaaSbaaKqbGeaacuaH4oqCgaqeaaqcfayabaGaaiikaiqbec8a WzaajaGaaiilaiqbeI7aXzaaraGaaiykaaaa@4069@  = n ( X ¯ π μ( θ ) ) σ 2 (π,θ) π 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba WaaOaaaeaacaWGUbGaaGjbVdqabaWaaeWaaeaadaWcaaqaaiqadIfa gaqeaaqaaiabec8aWbaacqGHsislcqaH8oqBcaGGOaGaeqiUde3aaS baaeaaaeqaaiaacMcaaiaawIcacaGLPaaaaeaadaGcaaqaamaalaaa baGaeq4Wdm3aaWbaaeqajuaibaGaaGOmaaaajuaGcaGGOaGaeqiWda NaaiilaiabeI7aXjaacMcaaeaacqaHapaCdaahaaqabKqbGeaacaaI YaaaaaaaaKqbagqaaaaaaaa@4F87@ .

Simulation Study

A simulation study is carried out to investigate the power of the two tests proposed in section 3.2. We generate 10000 samples of sizes 100 and 200 for different values of p , θ and truncation point t. Based on generated sample, the test statistics were calculated. Percentage of times the test statistics exceeds Z1-a/2 is computed, which is an estimate of power of the respective test. R programme is developed to find power of the test. The results for the case of θ0=2 and 4 , p=0.3, 0.4, 0.5, 0.6, 0.7, a=0.05 and truncation point t= 7 and 9 are presented in the Table 1 & Table 2.

π

θ

n=100

n=200

ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@

ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@

ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@

ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@

0.3

2.0

6.57

4.28

6.57

4.63

2.2

11.49

8.9

16.08

12.88

2.4

26.85

22.24

45.08

39.93

2.6

49.43

44.5

76.81

72.82

2.8

71.27

66.99

93.49

91.72

3

86.28

83.23

98.91

98.46

3.2

94.64

93.18

99.77

99.73

3.4

98.08

97.58

99.99

99.99

3.6

99.44

99.08

100

100

3.8

99.8

99.76

100

100

4

99.95

99.94

100

100

4.2

99.98

99.97

100

100

4.4

100

100

100

100

0.4

2

6.44

4.24

6.29

4.46

2.2

12.83

10.33

20.01

15.59

2.4

33.16

28.94

56.94

50.24

2.6

60.11

55.4

87.34

83.64

2.8

81.87

78.72

97.82

97

3

93.8

92.31

99.83

99.79

3.2

98.35

97.87

100

100

3.4

99.62

99.54

100

100

3.6

99.98

99.96

100

100

3.8

99.99

99.97

100

100

3.8

100

100

100

100

0.5

2

6.17

4.46

6.25

4.2

2.2

14.83

12.01

24.63

19.24

2.4

40.31

34.76

66.99

60.39

2.6

70.38

65.06

92.88

90.37

2.8

90.13

87.37

99.52

99.14

3

97.23

96.45

99.99

99.97

3.2

99.55

99.36

100

100

3.4

99.96

99.94

100

100

3.6

100

99.99

100

100

3.8

100

100

100

100

0.6

2

6.8

4.43

7.12

4.89

2.2

18.04

13.52

28.35

21.91

2.4

47.01

40.71

73.41

65.85

2.6

77.65

72.17

96.33

94.68

2.8

94.05

91.78

99.86

99.74

3

99.01

98.48

99.99

99.99

3.2

99.85

99.74

100

99.99

3.4

99.99

99.97

100

100

3.6

100

100

100

100

0.7

2

7.11

4.17

7.34

4.95

2.2

19.69

14.15

32.46

24.21

2.4

54.29

45.76

80.95

73.64

2.6

84.17

78.59

98.35

97.26

2.8

96.91

95.1

99.95

99.9

3

99.65

99.28

100

100

3.2

99.99

99.97

100

100

3.4

100

100

100

100

Table 1: Power (in %) of the test ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@ and ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@  for θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIWaaajuaGbeaaaaa@39D1@ =2. t=7, n=100 and 200, α=0.05

π

n=100

n=100

n=200

ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@

ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@

ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@

ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@

0.3

4

5.56

3.58

4.65

3.37

4.2

9.33

4.71

12.38

5.58

4.4

19.4

9.95

31.31

17.04

4.6

33.8

20.81

56.28

38.36

4.8

50.58

35.48

78.37

62.91

5

68.14

53.07

92

82.84

5.2

80.88

67.97

97.5

93.47

5.4

89.97

81.06

99.45

98.4

5.6

95.31

89.59

99.83

99.53

5.8

97.77

94.74

99.97

99.94

6

99.05

97.48

100

99.99

6.2

99.6

98.72

100

100

6.4

99.85

99.5

100

100

0.4

4

5.29

3.57

5.26

3.95

4.2

10.24

4.86

13.8

5.8

4.4

22.52

12.32

38.38

21.74

4.6

41.49

26.14

68.09

49.91

4.8

62.45

46.12

88.35

76.97

5

78.69

65.52

97.45

92.41

5.2

90.17

81.34

99.52

98.26

5.4

95.75

90.75

99.96

99.67

5.6

98.55

96.16

99.99

99.97

5.8

99.53

98.56

100

100

6

99.88

99.52

100

100

6.2

99.94

99.78

100

100

6.4

99.96

99.94

100

100

0.5

4

5.39

3.75

4.88

3.91

4.2

11.78

5.51

15.75

6.94

4.4

26.72

14.86

45.12

26.2

4.6

49.72

33.44

76.69

59.34

4.8

70.81

55

94.06

85.45

5

86.58

75.44

98.94

96.71

5.2

95.84

89.47

99.95

99.49

5.4

98.59

95.86

99.98

99.95

5.6

99.62

98.82

100

100

5.8

99.93

99.79

100

100

6

99.98

99.86

100

100

6.2

99.99

99.97

100

100

6.4

100

100

100

100

0.6

4

4.71

3.41

5.27

4.12

4.2

13.45

5.88

20.35

8.06

4.4

34.38

19.15

57.57

35.63

4.6

62.82

45.15

89.27

74.97

4.8

84.74

70.72

98.5

94.9

5

95.58

88.95

99.96

99.56

5.2

98.9

96.8

100

99.97

5.4

99.77

99.19

100

100

5.6

99.98

99.88

100

100

5.8

100

100

100

100

0.7

4

4.71

3.41

5.27

4.12

4.2

13.45

5.88

20.35

8.06

4.4

34.38

19.15

57.57

35.63

4.6

62.82

45.15

89.27

74.97

4.8

84.74

70.72

98.5

94.9

5

95.58

88.95

99.96

99.56

5.2

98.9

96.8

100

99.97

5.4

99.77

99.19

100

100

5.6

99.98

99.88

100

100

5.8

100

100

100

100

Table 2: Power (in %) of the test ψ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGinaaqcfayabaaaaa@39E2@ and ψ 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI8a5n aaBaaajuaibaGaaGynaaqcfayabaaaaa@39E3@ for θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIWaaajuaGbeaaaaa@39D1@ =4, t=9, n=100 and 200, a=0.05

From the simulation study reported in Table 1 & Table 2, we observe that

  1. The test based on full likelihood approach is better than the one based on conditional likelihood approach when θ is small. For large θ, both the tests are equally good.
  2. Probability of Type I error of the former test is more than that of later.
  3. Since for large values of θ both the tests are equally good. We recommend the use of conditional likelihood approach, when θ is large, from the computational point of view.
  4. If θ is large, proportion of zeros corresponding the Poisson distribution are relatively low. Hence these zeros can be ignored while making inference about θ. However, for smaller values of θ, such ignorance will have effect on inference of θ.

Illustrative Example

Let us consider the data of Traffic Accident Research given by Kuan et al. [5].

The data from the department of motor vehicles master driver license file

Traffic accidents 0

1

2

>3

Number of drivers

4499

766

136

21

From the data we see that there is excess number of zero counts and the frequency of X is greater than or equal to 3 is 21. Generally such data is modeled by Poisson distribution. But Poisson distribution does not fit well for this data. We fit the above data for ZIPD. In ZIPD there are two parameters π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ . In this problem n 0 =4499,n=5422 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gada WgaaqcfasaaiaaicdaaKqbagqaaiabg2da9iaaisdacaaI0aGaaGyo aiaaiMdacaGGSaGaaGPaVlaaywW7caWGUbGaeyypa0JaaGynaiaais dacaaIYaGaaGOmaaaa@45C2@  and estimated values of π ^ =0.5583 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0JaaGimaiaac6cacaaI1aGaaGynaiaaiIdacaaIZaaa aa@3DB5@  and θ ^ =0.3637 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaGaeyypa0JaaGimaiaac6cacaaIZaGaaGOnaiaaiodacaaI3aaa aa@3DAC@ . Using these values we fit the ZIPD for the above data. The calculated Chi-square value is 0.4043 and table value of X2(1, 0.05) is 3.841459 and the P-value is 0.5249

Same data is fitted to ZITPD truncated at 4 and above. The parameters are π ^ =0.5582 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0JaaGimaiaac6cacaaI1aGaaGynaiaaiIdacaaIYaaa aa@3DB4@  and θ ^ =0.3646 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaGaeyypa0JaaGimaiaac6cacaaIZaGaaGOnaiaaisdacaaI2aaa aa@3DAC@  The calculated Chi-square value is 0.4018 and table value of X2(1, 0.05) is 3.8415 and the P-value is 0.5262. If the same data is fitted to ZITPD truncated at 5 and above. The parameters are π ^ =0.5583 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0JaaGimaiaac6cacaaI1aGaaGynaiaaiIdacaaIZaaa aa@3DB5@  and θ ^ =0.3638 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeI7aXz aajaGaeyypa0JaaGimaiaac6cacaaIZaGaaGOnaiaaiodacaaI4aaa aa@3DAD@  . The calculated Chi-square value is 0.4012 and table value of X2(1, 0.05) is 3.8415 and the P-value is 0.5265. Here we prefer ZITPD to model the data.

References

  1. Gupta PL, Gupta RL, Tripathi RC (1995) Inflated Modified Power Series Distributions with Applications. Comm StatistTheory Meth 24(9): 2355-2374.
  2. Murat M, Szynal D (1998) Non-Zero-Inflated Modified Power Series Distributions. Commun Statist.Theory Meth 27(12): 3047-3064.
  3.  Patil MK, Shirke DT (2011) Tests for equality of inflation parameters of two zero-inflated power series distributions. CommunStatist Theory Meth 40(14): 2539 -2553.
  4. Patil MK, Shirke DT (2007) Testing parameter of the power series distribution of a zero-inflated power series model, Statistical Methodology4(4): 393-406.
  5. KuanJ, Peck RC, Janke MK (1991) Statistical Methods for Traffic Accidents Research, in proceeding of the 1990 Taipei Symposium in statistics, June 28-30, 1990, (Eds), by Min - Te Chao and Philip E Cheng Taipei, Institute of Statistical Science.
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