ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 2 Issue 6 - 2014
On Zero-Truncation of Poisson and Poisson-Lindley Distributions and Their Applications
Rama Shanker1*, Hagos Fesshaye2, Sujatha Selvaraj3 and Abrehe Yemane4
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
3Department of Banking and Finance, Jimma University, Ethiopia
4Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: June 23, 2015 | Published: July 22, 2015
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H, Selvaraj S, Yemane A (2015) On Zero-Truncation of Poisson and Poisson-Lindley Distributions and Their Applications. Biom Biostat Int J 2(6): 00045. DOI: 10.15406/bbij.2015.02.00045

Abstract

In this paper, a general expression for the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th factorial moment of zero-truncated Poisson-Lindley distribution (ZTPLD) has been obtained and hence the first four moments about origin has been given. A very simple and alternative method for finding moments of ZTPLD has also been suggested. The expression for the moment generating function of ZTPLD has been obtained. Both ZTPD (Zero-truncated Poisson distribution) and ZTPLD have been fitted using maximum likelihood estimate to a number of data- sets from demography, biological sciences and social sciences and it has been observed that in most cases ZTPLD gives much closer fits than ZTPD while in some cases ZTPD gives much closer fits than ZTPLD.

Keywords: Poisson-Lindley distribution; Zero-truncated distribution; Moments; Estimation of parameter; Goodness of fit

Abbreviations

ZTPLD: Zero-Truncated Poisson-Lindley Distribution; ZTPD: Zero-Truncated Poisson Distribution; PLD: Poisson-Lindley Distribution; PMF: Probability Mass Function; PDF: Probability Density Function; SBPD: Size-Biased Poisson Distribution; MLE: Maximum Likelihood Estimate; MOME: Method of Moment Estimate; MVUE: Minimum Variance Unbiased Estimation; SBPD: Size-Biased Poisson Distribution

Introduction

Zero-truncated distributions, in probability theory, are certain discrete distributions having support the set of positive integers. These distributions are applicable for the situations when the data to be modeled originate from a mechanism that generates data excluding zero-counts.

Let P 0 ( x;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzGeGaaGimaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaaaaa@3F94@  is the original distribution with support non negative positive integers. Then the zero-truncated version of P 0 ( x;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzGeGaaGimaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaaaaa@3F94@  with the support the set of positive integers is given by
P( x;θ )= P 0 ( x;θ ) 1 P 0 ( 0;θ ) ;x=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamiuaKqbaoaaBa aaleaajugibiaaicdaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaiaa cUdacqaH4oqCaOGaayjkaiaawMcaaaqaaKqzGeGaaGymaiabgkHiTi aadcfajuaGdaWgaaWcbaqcLbsacaaIWaaaleqaaKqbaoaabmaakeaa jugibiaaicdacaGG7aGaeqiUdehakiaawIcacaGLPaaaaaqcLbsaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaacUdacaWG4bGaeyypa0Ja aGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOlaiaac6caca GGUaaaaa@65C6@  (1.1)
The Poisson-Lindley distribution (PLD) with parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  and having probability mass function (p.m.f.)
P 0 ( x;θ )= θ 2 ( x+θ+2 ) ( θ+1 ) x+3 ;x=0,1,2,3,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzGeGaaGimaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOm aaaajuaGdaqadaGcbaqcLbsacaWG4bGaey4kaSIaeqiUdeNaey4kaS IaaGOmaaGccaGLOaGaayzkaaaabaqcfa4aaeWaaOqaaKqzGeGaeqiU deNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaaju gibiaadIhacqGHRaWkcaaIZaaaaaaacaaMc8UaaGPaVlaaykW7caaM c8Uaai4oaiaadIhacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGaaG OmaiaacYcacaaIZaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaaM c8UaaGPaVlabeI7aXjabg6da+iaaicdaaaa@6FEC@  (1.2)
has been introduced by Sankaran [1] to model count data. Recently, Shanker et al. [2] have done an extensive study on its applications to Biological Sciences and found that PLD provides a better fit than Poisson distribution to almost all biological science data. The PSD arises from the Poisson distribution when its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37AA@ follows Lindley distribution [3] with probability density function (p.d.f.).
g( λ;θ )= θ 2 θ+1 ( 1+λ ) e θλ λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGNb qcfa4aaeWaaOqaaKqzGeGaeq4UdWMaai4oaiabeI7aXbGccaGLOaGa ayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGda ahaaWcbeqaaKqzGeGaaGOmaaaaaOqaaKqzGeGaeqiUdeNaey4kaSIa aGymaaaajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIaeq4UdWgaki aawIcacaGLPaaajugibiaadwgajuaGdaahaaWcbeqaaKqzGeGaeyOe I0IaeqiUdeNaeq4UdWgaaiaaykW7caaMc8UaaGPaVlaaykW7cqaH7o aBcqGH+aGpcaaIWaGaaGPaVlaaykW7caGGSaGaeqiUdeNaeyOpa4Ja aGimaaaa@66CD@ (1.3)
Detailed study of Lindley distribution (1.3) has been done by Ghitany et al. [4] and shown that (1.3) is a better model than exponential distribution. Recently, Shanker et al. [2] showed that (1.3) is not always a better model than the exponential distribution for modeling lifetimes data. In fact, Shanker et al. [2] has done a very extensive and comparative study on modeling of lifetimes data using exponential and Lindley distributions and discussed various lifetimes examples to show the superiority of Lindley over exponential and that of exponential over Lindley distribution. The PLD has been extensively studied by Sankaran [1] and Ghitany & Mutairi [5] and they have discussed its various properties. The Lindley distribution and the PLD has been generalized by many researchers. Shanker et al. [6] obtained a two parameter Poisson-Lindley distribution by compounding Poisson distribution with a two parameter Lindley distribution introduced by Shanker et al. [7]. A quasi Poisson-Lindley distribution has been introduced by Shanker et al. [8] by compounding Poisson distribution with a quasi Lindley distribution introduced by Shanker et al. [9]. Shanker et al. [10] obtained a discrete two parameter Poisson-Lindley distribution by mixing Poisson distribution with a two parameter Lindley distribution for modeling waiting and survival times data introduced by Shanker et al. [11]. Further, Shanker et al. [12] obtained a new quasi Poisson-Lindley distribution by compounding Poisson distribution with a new quasi Lindley distribution introduced by Shanker et al. [13,14].

In this paper, the nature of zero-truncated Poisson distribution (ZTPD) and zero-truncated Poisson-Lindley distribution (ZTPLD) has been compared and studied using graphs for different values of their parameter. A general expression for the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th factorial moment of ZTPLD has been obtained and the first four moment about origin has been given. A very simple and easy method for finding moments of ZTPLD has been suggested. Both ZTPD and ZTPLD have been fitted to a number of data sets from different fields to study their goodness of fits and superiority of one over the other.

Zero-Truncated Poisson and Poisson-Lindley Distribution

Zero-truncated poisson distribution (ZTPD)
Using (1.1) and the p.m.f. of Poisson distribution, the p.m.f. of zero-truncated Poisson distribution (ZTPD) given by
  P 1 ( x;θ )= θ x ( e θ 1 )x! ;x=1,2,3,....,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaamiE aaaaaOqaaKqbaoaabmaakeaajugibiaadwgajuaGdaahaaWcbeqaaK qzGeGaeqiUdehaaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGa aGPaVlaadIhacaGGHaaaaiaaykW7caaMc8UaaGPaVlaacUdacaWG4b Gaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOl aiaac6cacaGGUaGaaiOlaiaacYcacaaMc8UaeqiUdeNaeyOpa4JaaG imaaaa@6692@  (2.1.1)
was obtained independently by Plackett [15] and David et al. [16] to model count data that structurally excludes zero counts. An extension of a truncated Poisson distribution and estimation in a truncated Poisson distribution when zeros and some ones are missing has been discussed by Cohen [17,18]. Tate et al. [19] has discussed minimum variance unbiased estimation (MVUE) for the truncated Poisson distribution.

Zero-truncated poisson-Lindley distribution (ZTPLD)
Using (1.1) and (1.2), the p.m.f. of zero-truncated Poisson- Lindley distribution (ZTPLD) given by
P 2 ( x;θ )= θ 2 θ 2 +3θ+1 x+θ+2 ( θ+1 ) x ;x=1,2,3,....,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzGeGaaGOmaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOm aaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikdaaa Gaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aaSaaaOqa aKqzGeGaamiEaiabgUcaRiabeI7aXjabgUcaRiaaikdaaOqaaKqbao aabmaakeaajugibiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMca aKqbaoaaCaaaleqabaqcLbsacaWG4baaaaaacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaacUdacaWG4bGaeyypa0JaaGymaiaacYcacaaI YaGaaiilaiaaiodacaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaacY cacaaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@7565@  (2.2.1)
was obtained by Ghitany et al. [20] to model count data for the missing zeros. It has been shown by Ghitany et al. [20] that ZTPLD can also arise from the size-biased Poisson distribution (SBPD) with p.m.f.
f( x|λ )= e λ λ x1 ( x1 )! ;x=1,2,3,...,λ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacYhacqaH7oaBaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamyzaKqbaoaaCa aaleqabaqcLbsacqGHsislcqaH7oaBaaGaeq4UdWwcfa4aaWbaaSqa beaajugibiaadIhacqGHsislcaaIXaaaaaGcbaqcfa4aaeWaaOqaaK qzGeGaamiEaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaaiyi aaaacaaMc8UaaGPaVlaaykW7caGG7aGaamiEaiabg2da9iaaigdaca GGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6cacaGGUaGaaiOlaiaa cYcacqaH7oaBcqGH+aGpcaaIWaaaaa@639F@ (2.2.2)
when its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBaaa@3839@ follows a distribution having p.d.f.
h( λ;θ )= θ 2 θ 2 +3θ+1 [ ( θ+1 )λ+( θ+2 ) ] e θλ ;λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaeq4UdWMaai4oaiabeI7aXbGccaGLOaGa ayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGda ahaaWcbeqaaKqzGeGaaGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWba aSqabeaajugibiaaikdaaaGaey4kaSIaaG4maiabeI7aXjabgUcaRi aaigdaaaqcfa4aamWaaOqaaKqbaoaabmaakeaajugibiabeI7aXjab gUcaRiaaigdaaOGaayjkaiaawMcaaKqzGeGaeq4UdWMaey4kaSscfa 4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzk aaaacaGLBbGaayzxaaqcLbsacaWGLbqcfa4aaWbaaSqabeaajugibi abgkHiTiabeI7aXjabeU7aSbaacaaMc8UaaGPaVlaacUdacqaH7oaB cqGH+aGpcaaIWaGaaiilaiaaykW7caaMc8UaeqiUdeNaeyOpa4JaaG imaaaa@7591@  (2.2.3)
Thus the p.m.f. of ZTPLD is obtained as
P( X=x )= 0 f( x|λ ) h( λ;θ )dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaeWaaOqaaKqzGeGaamiwaiabg2da9iaadIhaaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aa8qCaOqaaKqzGeGaamOzaKqbaoaabm aakeaajugibiaadIhacaGG8bGaeq4UdWgakiaawIcacaGLPaaaaSqa aKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYdGaeyyXICTaam iAaKqbaoaabmaakeaajugibiabeU7aSjaacUdacqaH4oqCaOGaayjk aiaawMcaaKqzGeGaamizaiabeU7aSbaa@59FC@
= 0 e λ λ x1 ( x1 )! θ 2 θ 2 +3θ+1 [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWdXbGcbaqcfa4aaSaaaOqaaKqzGeGaamyzaKqbaoaaCaaa leqabaqcLbsacqGHsislcqaH7oaBaaGaeq4UdWwcfa4aaWbaaSqabe aajugibiaadIhacqGHsislcaaIXaaaaaGcbaqcfa4aaeWaaOqaaKqz GeGaamiEaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaaiyiaa aaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYdGaeyyX ICDcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibi aaikdaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbsacaaI YaaaaiabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaIXaaaaKqbaoaadm aakeaajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaa wIcacaGLPaaajugibiabeU7aSjabgUcaRKqbaoaabmaakeaajugibi abeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaGaay5waiaaw2fa aKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbsacqGHsislcqaH4oqCcq aH7oaBaaGaamizaiabeU7aSbaa@7C24@  (2.2.4)
= θ 2 ( θ 2 +3θ+1 )( x1 )! 0 e ( θ+1 )λ [ ( θ+1 ) λ x +( θ+2 ) λ x1 ]dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa aGOmaaaaaOqaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaale qabaqcLbsacaaIYaaaaiabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaI XaaakiaawIcacaGLPaaajuaGdaqadaGcbaqcLbsacaWG4bGaeyOeI0 IaaGymaaGccaGLOaGaayzkaaqcLbsacaGGHaaaaKqbaoaapehakeaa jugibiaadwgajuaGdaahaaWcbeqaaKqzGeGaeyOeI0scfa4aaeWaaS qaaKqzGeGaeqiUdeNaey4kaSIaaGymaaWccaGLOaGaayzkaaqcLbsa cqaH7oaBaaaaleaajugibiaaicdaaSqaaKqzGeGaeyOhIukacqGHRi I8aiabgwSixNqbaoaadmaakeaajuaGdaqadaGcbaqcLbsacqaH4oqC cqGHRaWkcaaIXaaakiaawIcacaGLPaaajugibiabeU7aSLqbaoaaCa aaleqabaqcLbsacaWG4baaaiabgUcaRKqbaoaabmaakeaajugibiab eI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaKqzGeGaeq4UdWwcfa 4aaWbaaSqabeaajugibiaadIhacqGHsislcaaIXaaaaaGccaGLBbGa ayzxaaqcLbsacaWGKbGaeq4UdWgaaa@8090@
= θ 2 θ 2 +3θ+1 [ x ( θ+1 ) x + ( θ+2 ) ( θ+1 ) x ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa aGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaik daaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aamWa aOqaaKqbaoaalaaakeaajugibiaadIhaaOqaaKqbaoaabmaakeaaju gibiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaaCaaa leqabaqcLbsacaWG4baaaaaacqGHRaWkjuaGdaWcaaGcbaqcfa4aae WaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaa baqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaGccaGLOa Gaayzkaaqcfa4aaWbaaSqabeaajugibiaadIhaaaaaaaGccaGLBbGa ayzxaaaaaa@62D7@
= θ 2 θ 2 +3θ+1 x+θ+2 ( θ+1 ) x ;x=1,2,3,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa aGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaik daaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aaSaa aOqaaKqzGeGaamiEaiabgUcaRiabeI7aXjabgUcaRiaaikdaaOqaaK qbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaa wMcaaKqbaoaaCaaaleqabaqcLbsacaWG4baaaaaacaaMc8UaaGPaVl aaykW7caGG7aGaamiEaiabg2da9iaaigdacaGGSaGaaGOmaiaacYca caaIZaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacqaH4oqCcqGH+a GpcaaIWaaaaa@6672@
which is the p.m.f. of ZTPLD with parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ .
To study the nature and behaviors of ZTPD and ZTPLD for different values of its parameter, a number of graphs of their probability densities have been drawn and presented in Figure 1.

Moments and related measures

Moments of ZTPD
 The r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb aaaa@377C@ th factorial moment of the ZTPD (2.1.1) can be obtained as
  μ ( r ) =E[ X ( r ) ]= 1 e θ 1 x=1 x ( r ) θ x x! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcfa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGa ayzkaaaabeaajuaGdaahaaWcbeqaaKqzGeGamai4gkdiIcaacqGH9a qpcaWGfbqcfa4aamWaaOqaaKqzGeGaamiwaKqbaoaaCaaaleqabaqc fa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGaayzkaaaaaaGccaGLBb GaayzxaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaa jugibiaadwgajuaGdaahaaWcbeqaaKqzGeGaeqiUdehaaiabgkHiTi aaigdaaaqcfa4aaabCaOqaaKqzGeGaamiEaKqbaoaaCaaaleqabaqc fa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGaayzkaaaaaaqaaKqzGe GaamiEaiabg2da9iaaigdaaSqaaKqzGeGaeyOhIukacqGHris5aKqb aoaalaaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbsacaWG4b aaaaGcbaqcLbsacaWG4bGaaiyiaaaacaaMc8UaaGPaVdaa@6D63@ , where X ( r ) =X(X1)(X2)...(Xr+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb qcfa4aaWbaaSqabeaajuaGdaqadaWcbaqcLbsacaWGYbaaliaawIca caGLPaaaaaqcLbsacqGH9aqpcaWGybGaaiikaiaadIfacqGHsislca aIXaGaaiykaiaacIcacaWGybGaeyOeI0IaaGOmaiaacMcacaGGUaGa aiOlaiaac6cacaGGOaGaamiwaiabgkHiTiaadkhacqGHRaWkcaaIXa Gaaiykaaaa@4DCC@
  = θ r e θ 1 x=r θ xr ( xr )! = θ r e θ e θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa amOCaaaaaOqaaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbsacqaH4o qCaaGaeyOeI0IaaGymaaaajuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqz GeGaeqiUdexcfa4aaWbaaSqabeaajugibiaadIhacqGHsislcaWGYb aaaaGcbaqcfa4aaeWaaOqaaKqzGeGaamiEaiabgkHiTiaadkhaaOGa ayjkaiaawMcaaKqzGeGaaiyiaaaacqGH9aqpjuaGdaWcaaGcbaqcLb sacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaamOCaaaacaaMc8Uaamyz aKqbaoaaCaaaleqabaqcLbsacqaH4oqCaaaakeaajugibiaadwgaju aGdaahaaWcbeqaaKqzGeGaeqiUdehaaiabgkHiTiaaigdaaaaaleaa jugibiaadIhacqGH9aqpcaWGYbaaleaajugibiabg6HiLcGaeyyeIu oaaaa@6B0A@ (3.1.1)
 Substituting r=1,2,3,and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb Gaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaGPa VlaaykW7caqGHbGaaeOBaiaabsgacaaMc8UaaGPaVlaabsdaaaa@4665@ in (3.1.1), the first four factorial moments can be obtained and, therefore, using the relationship between factorial moments and moments about origin, the first four moments about origin of ZTPD (2.1.1) are obtained as
  μ 1 = θ e θ e θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajugibiabeI7aXj aaykW7caWGLbqcfa4aaWbaaSqabeaajugibiabeI7aXbaaaOqaaKqz GeGaamyzaKqbaoaaCaaaleqabaqcLbsacqaH4oqCaaGaeyOeI0IaaG ymaaaaaaa@4E0F@
  μ 2 = θ e θ e θ 1 ( θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIYaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajugibiabeI7aXj aaykW7caWGLbqcfa4aaWbaaSqabeaajugibiabeI7aXbaaaOqaaKqz GeGaamyzaKqbaoaaCaaaleqabaqcLbsacqaH4oqCaaGaeyOeI0IaaG ymaaaajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaa wIcacaGLPaaaaaa@541D@
  μ 3 = θ e θ e θ 1 ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIZaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajugibiabeI7aXj aaykW7caWGLbqcfa4aaWbaaSqabeaajugibiabeI7aXbaaaOqaaKqz GeGaamyzaKqbaoaaCaaaleqabaqcLbsacqaH4oqCaaGaeyOeI0IaaG ymaaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqz GeGaaGOmaaaacqGHRaWkcaaIZaGaeqiUdeNaey4kaSIaaGymaaGcca GLOaGaayzkaaaaaa@5979@
  μ 3 = θ e θ e θ 1 ( θ 3 +6 θ 2 +7θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIZaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajugibiabeI7aXj aaykW7caWGLbqcfa4aaWbaaSqabeaajugibiabeI7aXbaaaOqaaKqz GeGaamyzaKqbaoaaCaaaleqabaqcLbsacqaH4oqCaaGaeyOeI0IaaG ymaaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqz GeGaaG4maaaacqGHRaWkcaaI2aGaeqiUdexcfa4aaWbaaSqabeaaju gibiaaikdaaaGaey4kaSIaaG4naiabeI7aXjabgUcaRiaaigdaaOGa ayjkaiaawMcaaaaa@5EDC@
Generating Function: The probability generating function of the ZTPD (2.1.1) is obtained as
  P X ( t )=E( t X )= 1 e θ 1 x=1 ( θt ) x x! = 1 e θ 1 [ x=0 ( θt ) x x! 1 ]= e θt 1 e θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzGeGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabg2da9iaadweajuaGdaqada GcbaqcLbsacaWG0bqcfa4aaWbaaSqabeaajugibiaadIfaaaaakiaa wIcacaGLPaaajugibiabg2da9KqbaoaalaaakeaajugibiaaigdaaO qaaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbsacqaH4oqCaaGaeyOe I0IaaGymaaaajuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqbaoaabmaake aajugibiabeI7aXjaadshaaOGaayjkaiaawMcaaKqbaoaaCaaaleqa baqcLbsacaWG4baaaaGcbaqcLbsacaWG4bGaaiyiaaaaaSqaaKqzGe GaamiEaiabg2da9iaaigdaaSqaaKqzGeGaeyOhIukacqGHris5aiab g2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamyzaKqbao aaCaaaleqabaqcLbsacqaH4oqCaaGaeyOeI0IaaGymaaaajuaGdaWa daGcbaqcfa4aaabCaOqaaKqbaoaalaaakeaajuaGdaqadaGcbaqcLb sacqaH4oqCcaWG0baakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqz GeGaamiEaaaaaOqaaKqzGeGaamiEaiaacgcaaaGaeyOeI0IaaGymaa WcbaqcLbsacaWG4bGaeyypa0JaaGimaaWcbaqcLbsacqGHEisPaiab ggHiLdaakiaawUfacaGLDbaajugibiabg2da9Kqbaoaalaaakeaaju gibiaadwgajuaGdaahaaWcbeqaaKqzGeGaeqiUdeNaamiDaaaacqGH sislcaaIXaaakeaajugibiaadwgajuaGdaahaaWcbeqaaKqzGeGaeq iUdehaaiabgkHiTiaaigdaaaaaaa@9179@
The moment generating function of the ZTPD (2.1.1) is thus given by
  M X ( t )=E( e tX )= e θ e t 1 e θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzGeGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabg2da9iaadweajuaGdaqada GcbaqcLbsacaWGLbqcfa4aaWbaaSqabeaajugibiaadshacaWGybaa aaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsaca WGLbqcfa4aaWbaaSqabeaajugibiabeI7aXjaaykW7caWGLbqcfa4a aWbaaWqabeaajugibiaadshaaaaaaiabgkHiTiaaigdaaOqaaKqzGe GaamyzaKqbaoaaCaaaleqabaqcLbsacqaH4oqCaaGaeyOeI0IaaGym aaaaaaa@59A0@

Moments of ZTPLD
 The r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb aaaa@377C@ th factorial moment of the ZTPLD (2.2.1) can be obtained as
μ ( r ) =E[ X ( r ) |λ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcfa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGa ayzkaaaabeaajuaGdaahaaWcbeqaaKqzGeGamai4gkdiIcaacqGH9a qpcaWGfbqcfa4aamWaaOqaaKqzGeGaamiwaKqbaoaaCaaaleqabaqc fa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGaayzkaaaaaKqzGeGaai iFaiabeU7aSbGccaGLBbGaayzxaaaaaa@4E5B@ , where X ( r ) =X(X1)(X2)...(Xr+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb qcfa4aaWbaaSqabeaajuaGdaqadaWcbaqcLbsacaWGYbaaliaawIca caGLPaaaaaqcLbsacqGH9aqpcaWGybGaaiikaiaadIfacqGHsislca aIXaGaaiykaiaacIcacaWGybGaeyOeI0IaaGOmaiaacMcacaGGUaGa aiOlaiaac6cacaGGOaGaamiwaiabgkHiTiaadkhacqGHRaWkcaaIXa Gaaiykaaaa@4DCC@
Using (2.2.4), we get
μ ( r ) = 0 [ x=1 x ( r ) e λ λ x1 ( x1 )! ] θ 2 θ 2 +3θ+1 [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcfa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGa ayzkaaaabeaajuaGdaahaaWcbeqaaKqzGeGamai4gkdiIcaacqGH9a qpjuaGdaWdXbGcbaqcfa4aamWaaOqaaKqbaoaaqahakeaajugibiaa dIhajuaGdaahaaWcbeqaaKqbaoaabmaaleaajugibiaadkhaaSGaay jkaiaawMcaaaaajuaGdaWcaaGcbaqcLbsacaWGLbqcfa4aaWbaaSqa beaajugibiabgkHiTiabeU7aSbaacqaH7oaBjuaGdaahaaWcbeqaaK qzGeGaamiEaiabgkHiTiaaigdaaaaakeaajuaGdaqadaGcbaqcLbsa caWG4bGaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcLbsacaGGHaaaaa WcbaqcLbsacaWG4bGaeyypa0JaaGymaaWcbaqcLbsacqGHEisPaiab ggHiLdaakiaawUfacaGLDbaaaSqaaKqzGeGaaGimaaWcbaqcLbsacq GHEisPaiabgUIiYdGaeyyXICDcfa4aaSaaaOqaaKqzGeGaeqiUdexc fa4aaWbaaSqabeaajugibiaaikdaaaaakeaajugibiabeI7aXLqbao aaCaaaleqabaqcLbsacaaIYaaaaiabgUcaRiaaiodacqaH4oqCcqGH RaWkcaaIXaaaaKqbaoaadmaakeaajuaGdaqadaGcbaqcLbsacqaH4o qCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajugibiabeU7aSjabgUca RKqbaoaabmaakeaajugibiabeI7aXjabgUcaRiaaikdaaOGaayjkai aawMcaaaGaay5waiaaw2faaKqzGeGaamyzaKqbaoaaCaaaleqabaqc LbsacqGHsislcqaH4oqCcaaMc8Uaeq4UdWgaaiaadsgacqaH7oaBaa a@98B7@
= 0 [ λ r1 x=r x e λ λ xr ( xr )! ] θ 2 θ 2 +3θ+1 [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWdXbGcbaqcfa4aamWaaOqaaKqzGeGaeq4UdWwcfa4aaWba aSqabeaajugibiaadkhacqGHsislcaaIXaaaaKqbaoaaqahakeaaju gibiaadIhajuaGdaWcaaGcbaqcLbsacaWGLbqcfa4aaWbaaSqabeaa jugibiabgkHiTiabeU7aSbaacqaH7oaBjuaGdaahaaWcbeqaaKqzGe GaamiEaiabgkHiTiaadkhaaaaakeaajuaGdaqadaGcbaqcLbsacaWG 4bGaeyOeI0IaamOCaaGccaGLOaGaayzkaaqcLbsacaGGHaaaaaWcba qcLbsacaWG4bGaeyypa0JaamOCaaWcbaqcLbsacqGHEisPaiabggHi LdaakiaawUfacaGLDbaaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEi sPaiabgUIiYdGaeyyXICDcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4a aWbaaSqabeaajugibiaaikdaaaaakeaajugibiabeI7aXLqbaoaaCa aaleqabaqcLbsacaaIYaaaaiabgUcaRiaaiodacqaH4oqCcqGHRaWk caaIXaaaaKqbaoaadmaakeaajuaGdaqadaGcbaqcLbsacqaH4oqCcq GHRaWkcaaIXaaakiaawIcacaGLPaaajugibiabeU7aSjabgUcaRKqb aoaabmaakeaajugibiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawM caaaGaay5waiaaw2faaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbsa cqGHsislcqaH4oqCcaaMc8Uaeq4UdWgaaiaadsgacqaH7oaBaaa@90D4@
Taking x+r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b Gaey4kaSIaamOCaaaa@395B@ in place of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aaaa@3782@ , we get
μ ( r ) = 0 λ r1 [ x=0 ( x+r ) e λ λ x x! ] θ 2 θ 2 +3θ+1 [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcfa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGa ayzkaaaabeaajuaGdaahaaWcbeqaaKqzGeGamai4gkdiIcaacqGH9a qpjuaGdaWdXbGcbaqcLbsacqaH7oaBjuaGdaahaaWcbeqaaKqzGeGa amOCaiabgkHiTiaaigdaaaqcfa4aamWaaOqaaKqbaoaaqahakeaaju aGdaqadaGcbaqcLbsacaWG4bGaey4kaSIaamOCaaGccaGLOaGaayzk aaqcfa4aaSaaaOqaaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbsacq GHsislcqaH7oaBaaGaeq4UdWwcfa4aaWbaaSqabeaajugibiaadIha aaaakeaajugibiaadIhacaGGHaaaaaWcbaqcLbsacaWG4bGaeyypa0 JaaGimaaWcbaqcLbsacqGHEisPaiabggHiLdaakiaawUfacaGLDbaa aSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYdGaeyyXIC Dcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaa ikdaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbsacaaIYa aaaiabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaIXaaaaKqbaoaadmaa keaajuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawI cacaGLPaaajugibiabeU7aSjabgUcaRKqbaoaabmaakeaajugibiab eI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaGaay5waiaaw2faaK qzGeGaamyzaKqbaoaaCaaaleqabaqcLbsacqGHsislcqaH4oqCcaaM c8Uaeq4UdWgaaiaadsgacqaH7oaBaaa@986E@
It is obvious that the expression within the bracket is λ+r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBcqGHRaWkcaWGYbaaaa@3A12@ and hence, we have
μ ( r ) = θ 2 θ 2 +3θ+1 0 λ r1 ( λ+r ) [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcfa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGa ayzkaaaabeaajuaGdaahaaWcbeqaaKqzGeGamai4gkdiIcaacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa aGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaik daaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aa8qC aOqaaKqzGeGaeq4UdWwcfa4aaWbaaSqabeaajugibiaadkhacqGHsi slcaaIXaaaaKqbaoaabmaakeaajugibiabeU7aSjabgUcaRiaadkha aOGaayjkaiaawMcaaaWcbaqcLbsacaaIWaaaleaajugibiabg6HiLc Gaey4kIipacqGHflY1juaGdaWadaGcbaqcfa4aaeWaaOqaaKqzGeGa eqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcLbsacqaH7oaBcq GHRaWkjuaGdaqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIYaaakiaa wIcacaGLPaaaaiaawUfacaGLDbaajugibiaadwgajuaGdaahaaWcbe qaaKqzGeGaeyOeI0IaeqiUdeNaaGPaVlabeU7aSbaacaWGKbGaeq4U dWgaaa@822B@
Using gamma integral and little algebraic simplification, we get finally, a general expression for the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb aaaa@377C@ th factorial moment of the ZTPLD (2.2.1) as
μ ( r ) = r! ( θ+1 ) 2 ( r+θ+1 ) θ r ( θ 2 +3θ+1 ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcfa4aaeWaaSqaaKqzGeGaamOCaaWccaGLOaGa ayzkaaaabeaajuaGdaahaaWcbeqaaKqzGeGamai4gkdiIcaacqGH9a qpjuaGdaWcaaGcbaqcLbsacaWGYbGaaiyiaiaaykW7juaGdaqadaGc baqcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGda ahaaWcbeqaaKqzGeGaaGOmaaaajuaGdaqadaGcbaqcLbsacaWGYbGa ey4kaSIaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaaabaqcLb sacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaamOCaaaacaaMc8Ecfa4a aeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikdaaa Gaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMca aaaajugibiaaykW7caaMc8Uaai4oaiaadkhacqGH9aqpcaaIXaGaai ilaiaaikdacaGGSaGaaG4maiaacYcacaGGUaGaaiOlaiaac6caaaa@746B@ (3.2.1)
Substituting r=1,2,3,and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb Gaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaGPa VlaabggacaqGUbGaaeizaiaaykW7caaI0aaaaa@4356@  in (3.2.1), the first four factorial moment can be obtained and then using the relationship between factorial moments and moments about origin, the first four moments about origin of the ZTPLD (2.2.1) are given by
μ 1 = ( θ+1 ) 2 ( θ+2 ) θ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajuaGdaqadaGcba qcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaah aaWcbeqaaKqzGeGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4oqCcq GHRaWkcaaIYaaakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbaoaa bmaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbsacaaIYaaaai abgUcaRiaaiodacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaa aaaaaa@5BF8@
μ 2 = ( θ+1 ) 2 ( θ 2 +4θ+6 ) θ 2 ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIYaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajuaGdaqadaGcba qcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaah aaWcbeqaaKqzGeGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcaaI0aGaeqiUdeNa ey4kaSIaaGOnaaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaGda ahaaWcbeqaaKqzGeGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4oqC juaGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcaaIZaGaeqiUde Naey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa@635F@  
μ 3 = ( θ+1 ) 2 ( θ 3 +8 θ 2 +24θ+24 ) θ 3 ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaIZaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajuaGdaqadaGcba qcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaah aaWcbeqaaKqzGeGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqaaKqzGeGaaG4maaaacqGHRaWkcaaI4aGaeqiUdexc fa4aaWbaaSqabeaajugibiaaikdaaaGaey4kaSIaaGOmaiaaisdacq aH4oqCcqGHRaWkcaaIYaGaaGinaaGccaGLOaGaayzkaaaabaqcLbsa cqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaG4maaaajuaGdaqadaGcba qcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWk caaIZaGaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa@6A38@
μ 4 = ( θ+1 ) 2 ( θ 4 +16 θ 3 +78 θ 2 +168θ+120 ) θ 4 ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbsacaaI0aaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9KqbaoaalaaakeaajuaGdaqadaGcba qcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaah aaWcbeqaaKqzGeGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqaaKqzGeGaaGinaaaacqGHRaWkcaaIXaGaaGOnaiab eI7aXLqbaoaaCaaaleqabaqcLbsacaaIZaaaaiabgUcaRiaaiEdaca aI4aGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikdaaaGaey4kaSIa aGymaiaaiAdacaaI4aGaeqiUdeNaey4kaSIaaGymaiaaikdacaaIWa aakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbaoaaCaaaleqabaqc LbsacaaI0aaaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaale qabaqcLbsacaaIYaaaaiabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaI XaaakiaawIcacaGLPaaaaaaaaa@7290@
Generating function: The probability generating function of the ZTPLD (2.2.1) is obtained as
  P X ( t )=E( t X )= θ 2 θ 2 +3θ+1 x=1 t x x+θ+2 ( θ+1 ) x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb qcfa4aaSbaaSqaaKqzGeGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabg2da9iaadweajuaGdaqada GcbaqcLbsacaWG0bqcfa4aaWbaaSqabeaajugibiaadIfaaaaakiaa wIcacaGLPaaajugibiabg2da9KqbaoaalaaakeaajugibiabeI7aXL qbaoaaCaaaleqabaqcLbsacaaIYaaaaaGcbaqcLbsacqaH4oqCjuaG daahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcaaIZaGaeqiUdeNaey 4kaSIaaGymaaaajuaGdaaeWbGcbaqcLbsacaWG0bqcfa4aaWbaaSqa beaajugibiaadIhaaaaaleaajugibiaadIhacqGH9aqpcaaIXaaale aajugibiabg6HiLcGaeyyeIuoajuaGdaWcaaGcbaqcLbsacaWG4bGa ey4kaSIaeqiUdeNaey4kaSIaaGOmaaGcbaqcfa4aaeWaaOqaaKqzGe GaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqa beaajugibiaadIhaaaaaaaaa@7002@
= θ 2 θ 2 +3θ+1 [ x=1 x ( t θ+1 ) x +( θ+2 ) x=1 ( t θ+1 ) x ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa aGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaik daaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aamWa aOqaaKqbaoaaqahakeaajugibiaadIhajuaGdaqadaGcbaqcfa4aaS aaaOqaaKqzGeGaamiDaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaa aaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugibiaadIhaaaGaey 4kaSscfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGOmaaGccaGL OaGaayzkaaqcfa4aaabCaOqaaKqbaoaabmaakeaajuaGdaWcaaGcba qcLbsacaWG0baakeaajugibiabeI7aXjabgUcaRiaaigdaaaaakiaa wIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGaamiEaaaaaSqaaKqzGe GaamiEaiabg2da9iaaigdaaSqaaKqzGeGaeyOhIukacqGHris5aaWc baqcLbsacaWG4bGaeyypa0JaaGymaaWcbaqcLbsacqGHEisPaiabgg HiLdaakiaawUfacaGLDbaaaaa@763B@
  = θ 2 t θ 2 +3θ+1 [ θ+1 ( θ+1t ) 2 + θ+2 θ+1t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa aGOmaaaacaaMc8UaamiDaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbe qaaKqzGeGaaGOmaaaacqGHRaWkcaaIZaGaeqiUdeNaey4kaSIaaGym aaaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqzGeGaeqiUdeNaey4kaS IaaGymaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGym aiabgkHiTiaadshaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLb sacaaIYaaaaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqaH4oqCcqGH RaWkcaaIYaaakeaajugibiabeI7aXjabgUcaRiaaigdacqGHsislca WG0baaaaGccaGLBbGaayzxaaaaaa@64A9@
The moment generating function of the ZTPLD (2.2.1) is thus given by
M X ( t )=E( e tX )= θ 2 e t θ 2 +3θ+1 [ θ+1 ( θ+1 e t ) 2 + θ+2 θ+1 e t ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzGeGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabg2da9iaadweajuaGdaqada GcbaqcLbsacaWGLbqcfa4aaWbaaSqabeaajugibiaadshacaaMc8Ua amiwaaaaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikdaaaGaaGPaVlaa dwgajuaGdaahaaWcbeqaaKqzGeGaamiDaaaaaOqaaKqzGeGaeqiUde xcfa4aaWbaaSqabeaajugibiaaikdaaaGaey4kaSIaaG4maiabeI7a XjabgUcaRiaaigdaaaqcfa4aamWaaOqaaKqbaoaalaaakeaajugibi abeI7aXjabgUcaRiaaigdaaOqaaKqbaoaabmaakeaajugibiabeI7a XjabgUcaRiaaigdacqGHsislcaWGLbqcfa4aaWbaaSqabeaajugibi aadshaaaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzGeGaaGOm aaaaaaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqiUdeNaey4kaSIaaG OmaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaGaeyOeI0IaamyzaKqb aoaaCaaaleqabaqcLbsacaWG0baaaaaaaOGaay5waiaaw2faaaaa@7D38@

A Simple method of finding moments of ZTPLD

Using (2.2.4), the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb aaaa@377C@ th moment about origin of ZTPLD (2.2.1) can be obtained as
μ r =E[ E( X r |λ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH8o qBgaqbaKqbaoaaBaaaleaajugibiaadkhaaSqabaqcLbsacqGH9aqp caWGfbqcfa4aamWaaOqaaKqzGeGaamyraKqbaoaabmaakeaajugibi aadIfajuaGdaahaaWcbeqaaKqzGeGaamOCaaaacaGG8bGaeq4UdWga kiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@4960@  
= θ 2 θ 2 +3θ+1 0 [ x=1 x r e λ λ x1 ( x1 )! ] [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGa aGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaik daaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aa8qC aOqaaKqbaoaadmaakeaajuaGdaaeWbGcbaqcLbsacaWG4bqcfa4aaW baaSqabeaajugibiaadkhaaaaaleaajugibiaadIhacqGH9aqpcaaI Xaaaleaajugibiabg6HiLcGaeyyeIuoajuaGdaWcaaGcbaqcLbsaca WGLbqcfa4aaWbaaSqabeaajugibiabgkHiTiabeU7aSbaacqaH7oaB juaGdaahaaWcbeqaaKqzGeGaamiEaiabgkHiTiaaigdaaaaakeaaju aGdaqadaGcbaqcLbsacaWG4bGaeyOeI0IaaGymaaGccaGLOaGaayzk aaqcLbsacaGGHaaaaaGccaGLBbGaayzxaaaaleaajugibiaaicdaaS qaaKqzGeGaeyOhIukacqGHRiI8aiabgwSixNqbaoaadmaakeaajuaG daqadaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPa aajugibiabeU7aSjabgUcaRKqbaoaabmaakeaajugibiabeI7aXjab gUcaRiaaikdaaOGaayjkaiaawMcaaaGaay5waiaaw2faaKqzGeGaam yzaKqbaoaaCaaaleqabaqcLbsacqGHsislcqaH4oqCcaaMc8Uaeq4U dWgaaiaadsgacqaH7oaBaaa@8C35@  (4.1)
It is obvious that the expression under the bracket in (4.1) is the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb aaaa@377C@ th moment about origin of the SBPD. Taking r=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb Gaeyypa0JaaGymaaaa@393D@  in (4.1) and using the first moment about origin of the SBPD, the first moment about origin of the ZTPLD (2.2.1) is obtained as
μ 1 = θ 2 θ 2 +3θ+1 0 ( λ+1 ) [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH8o qBgaqbaKqbaoaaBaaaleaajugibiaaigdaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaG OmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikda aaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aa8qCaO qaaKqbaoaabmaakeaajugibiabeU7aSjabgUcaRiaaigdaaOGaayjk aiaawMcaaaWcbaqcLbsacaaIWaaaleaajugibiabg6HiLcGaey4kIi pajuaGdaWadaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIa aGymaaGccaGLOaGaayzkaaqcLbsacqaH7oaBcqGHRaWkjuaGdaqada GcbaqcLbsacqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaaaiaa wUfacaGLDbaajugibiaadwgajuaGdaahaaWcbeqaaKqzGeGaeyOeI0 IaeqiUdeNaaGPaVlabeU7aSbaacaWGKbGaeq4UdWgaaa@7384@   = ( θ+1 ) 2 ( θ+2 ) θ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIa aGymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugibiaaikdaaa qcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGa ayzkaaaabaqcLbsacqaH4oqCjuaGdaqadaGcbaqcLbsacqaH4oqCju aGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcaaIZaGaeqiUdeNa ey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa@5401@  (4.2)
Again taking r=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb Gaeyypa0JaaGOmaaaa@393E@  in (4.1) and using the second moment about origin of the SBPD, the second moment about origin of the ZTPLD (2.2.1) is obtained as
μ 2 = θ 2 θ 2 +3θ+1 0 ( λ 2 +3λ+1 ) [ ( θ+1 )λ+( θ+2 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH8o qBgaqbaKqbaoaaBaaaleaajugibiaaikdaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaG OmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikda aaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaaqcfa4aa8qCaO qaaKqbaoaabmaakeaajugibiabeU7aSLqbaoaaCaaaleqabaqcLbsa caaIYaaaaiabgUcaRiaaiodacqaH7oaBcqGHRaWkcaaIXaaakiaawI cacaGLPaaaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIi Ydqcfa4aamWaaOqaaKqbaoaabmaakeaajugibiabeI7aXjabgUcaRi aaigdaaOGaayjkaiaawMcaaKqzGeGaeq4UdWMaey4kaSscfa4aaeWa aOqaaKqzGeGaeqiUdeNaey4kaSIaaGOmaaGccaGLOaGaayzkaaaaca GLBbGaayzxaaqcLbsacaWGLbqcfa4aaWbaaSqabeaajugibiabgkHi TiabeI7aXjaaykW7cqaH7oaBaaGaamizaiabeU7aSbaa@78DE@
= ( θ+1 ) 2 ( θ 2 +4θ+6 ) θ 2 ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIa aGymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugibiaaikdaaa qcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaa ikdaaaGaey4kaSIaaGinaiabeI7aXjabgUcaRiaaiAdaaOGaayjkai aawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaikda aaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibi aaikdaaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaigdaaOGaayjk aiaawMcaaaaaaaa@5B67@ (4.3)
Similarly, taking r=3and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb Gaeyypa0JaaG4maiaaykW7caqGHbGaaeOBaiaabsgacaaMc8UaaGin aaaa@3FCF@ in (4.1) and using the respective moments of SBPD, we get finally, after a little simplification, the third and the fourth moments about origin of the ZTPLD (2.2.1) as
μ 3 = ( θ+1 ) 2 ( θ 3 +8 θ 2 +24θ+24 ) θ 3 ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH8o qBgaqbaKqbaoaaBaaaleaajugibiaaiodaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaG ymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugibiaaikdaaaqc fa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaaio daaaGaey4kaSIaaGioaiabeI7aXLqbaoaaCaaaleqabaqcLbsacaaI YaaaaiabgUcaRiaaikdacaaI0aGaeqiUdeNaey4kaSIaaGOmaiaais daaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaa jugibiaaiodaaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaS qabeaajugibiaaikdaaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaa igdaaOGaayjkaiaawMcaaaaaaaa@66A1@ (4.4)
μ 4 = ( θ+1 ) 2 ( θ 4 +16 θ 3 +78 θ 2 +168θ+120 ) θ 4 ( θ 2 +3θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH8o qBgaqbaKqbaoaaBaaaleaajugibiaaisdaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqiUdeNaey4kaSIaaG ymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugibiaaikdaaaqc fa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugibiaais daaaGaey4kaSIaaGymaiaaiAdacqaH4oqCjuaGdaahaaWcbeqaaKqz GeGaaG4maaaacqGHRaWkcaaI3aGaaGioaiabeI7aXLqbaoaaCaaale qabaqcLbsacaaIYaaaaiabgUcaRiaaigdacaaI2aGaaGioaiabeI7a XjabgUcaRiaaigdacaaIYaGaaGimaaGccaGLOaGaayzkaaaabaqcLb sacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGinaaaajuaGdaqadaGc baqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRa WkcaaIZaGaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa @6EF9@ (4.5)

Estimation of parameter

Estimation of Parameter of ZTPD
Maximum Likelihood Estimate (MLE): Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiaacYcacaWG4bqc fa4aaSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaacYcacaaMc8Uaai Olaiaac6cacaGGUaGaaGPaVlaacYcacaWG4bqcfa4aaSbaaSqaaKqz GeGaamOBaaWcbeaaaaa@483C@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@ from the ZTPD (2.1.1). The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ of ZTPD (2.1.1) is given by the solution of the following non linear equation.
  e θ ( x ¯ θ ) x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb qcfa4aaWbaaSqabeaajugibiabeI7aXbaajuaGdaqadaGcbaqcLbsa ceWG4bGbaebacqGHsislcqaH4oqCaOGaayjkaiaawMcaaKqzGeGaey OeI0IabmiEayaaraGaeyypa0JaaGimaaaa@4532@ , where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean
Method of Moment Estimate (MOME): Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiaacYcacaWG4bqc fa4aaSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaacYcacaaMc8Uaai Olaiaac6cacaGGUaGaaGPaVlaacYcacaWG4bqcfa4aaSbaaSqaaKqz GeGaamOBaaWcbeaaaaa@483C@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@  from the ZTPD (2.1.1). Equating the first population moment about origin to the corresponding sample moment, the MOME θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aaaaa@37BB@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  of ZTPD (2.1.1) is the solution of the following non linear equation.
  e θ ( x ¯ θ ) x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGLb qcfa4aaWbaaSqabeaajugibiabeI7aXbaajuaGdaqadaGcbaqcLbsa ceWG4bGbaebacqGHsislcqaH4oqCaOGaayjkaiaawMcaaKqzGeGaey OeI0IabmiEayaaraGaeyypa0JaaGimaaaa@4532@ , where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b Gbaebaaaa@379A@ is the sample mean
Thus both MLE and MOME give the same estimate of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  of ZTPD (2.1.1).

Figure 1: Graph of probability functions of ZTPD and ZTPLD for different values of their parameter. The left hand side graphs are for ZTPD and right hand side graphs are for ZTPLD.

Estimation of Parameter of ZTPLD
Maximum Likelihood Estimate (MLE): Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiaacYcacaWG4bqc fa4aaSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaacYcacaGGUaGaai Olaiaac6cacaGGSaGaamiEaKqbaoaaBaaaleaajugibiaad6gaaSqa baaaaa@4526@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ from the ZTPLD (2.2.1) and let f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzGeGaamiEaaWcbeaaaaa@39C1@ be the observed frequency in the sample corresponding to X=x(x=1,2,3,...,k) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb Gaeyypa0JaamiEaiaacIcacaWG4bGaeyypa0JaaGymaiaacYcacaaI YaGaaiilaiaaiodacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadU gacaGGPaaaaa@44BB@ such that x=1 k f x =n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCaO qaaKqzGeGaamOzaKqbaoaaBaaaleaajugibiaadIhaaSqabaaabaqc LbsacaWG4bGaeyypa0JaaGymaaWcbaqcLbsacaWGRbaacqGHris5ai abg2da9iaad6gaaaa@4353@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@  is the largest observed value having non-zero frequency. The likelihood function L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb aaaa@3756@ of the ZTPLD (2.2.1) is given by
L= ( θ 2 θ 2 +3θ+1 ) n 1 ( θ+1 ) x=1 k x f x x=1 k ( x+θ+2 ) f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0tcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiabeI7aXLqb aoaaCaaaleqabaqcLbsacaaIYaaaaaGcbaqcLbsacqaH4oqCjuaGda ahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcaaIZaGaeqiUdeNaey4k aSIaaGymaaaaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLbsaca WGUbaaaKqbaoaalaaakeaajugibiaaigdaaOqaaKqbaoaabmaakeaa jugibiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaaCa aaleqabaqcfa4aaabCaSqaaKqzGeGaamiEaiaadAgajuaGdaWgaaad baqcLbsacaWG4baameqaaaqaaKqzGeGaamiEaiabg2da9iaaigdaaW qaaKqzGeGaam4AaaGaeyyeIuoaaaaaaKqbaoaarahakeaajuaGdaqa daGcbaqcLbsacaWG4bGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaGcca GLOaGaayzkaaaaleaajugibiaadIhacqGH9aqpcaaIXaaaleaajugi biaadUgaaiabg+Givdqcfa4aaWbaaSqabeaajugibiaadAgajuaGda WgaaadbaqcLbsacaWG4baameqaaaaaaaa@742F@  (5.1.1)
The log likelihood function is given by
logL=nlog( θ 2 θ 2 +3θ+1 ) x=1 k x f x log( θ+1 ) + x=1 k f x log( x+θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb Gaai4BaiaacEgacaWGmbGaeyypa0JaamOBaiGacYgacaGGVbGaai4z aKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaa WcbeqaaKqzGeGaaGOmaaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqa beaajugibiaaikdaaaGaey4kaSIaaG4maiabeI7aXjabgUcaRiaaig daaaaakiaawIcacaGLPaaajugibiabgkHiTKqbaoaaqahakeaajugi biaadIhacaaMc8UaamOzaKqbaoaaBaaaleaajugibiaadIhaaSqaba qcLbsaciGGSbGaai4BaiaacEgajuaGdaqadaGcbaqcLbsacqaH4oqC cqGHRaWkcaaIXaaakiaawIcacaGLPaaaaSqaaKqzGeGaamiEaiabg2 da9iaaigdaaSqaaKqzGeGaam4AaaGaeyyeIuoacqGHRaWkjuaGdaae WbGcbaqcLbsacaWGMbqcfa4aaSbaaSqaaKqzGeGaamiEaaWcbeaaju gibiGacYgacaGGVbGaai4zaKqbaoaabmaakeaajugibiaadIhacqGH RaWkcqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaaaSqaaKqzGe GaamiEaiabg2da9iaaigdaaSqaaKqzGeGaam4AaaGaeyyeIuoaaaa@7FD0@
and the log likelihood equation is thus obtained as
dlogL dθ = 2n θ n( 2θ+3 ) θ 2 +3θ+1 n x ¯ θ+1 + x=1 k f x x+θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGVbGaai4zaiaadYeaaOqaaKqzGeGa amizaiabeI7aXbaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIYaGaam OBaaGcbaqcLbsacqaH4oqCaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGa amOBaKqbaoaabmaakeaajugibiaaikdacqaH4oqCcqGHRaWkcaaIZa aakiaawIcacaGLPaaaaeaajugibiabeI7aXLqbaoaaCaaaleqabaqc LbsacaaIYaaaaiabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaIXaaaai abgkHiTKqbaoaalaaakeaajugibiaad6gacaaMc8UabmiEayaaraaa keaajugibiabeI7aXjabgUcaRiaaigdaaaGaey4kaSscfa4aaabCaO qaaKqbaoaalaaakeaajugibiaadAgajuaGdaWgaaWcbaqcLbsacaWG 4baaleqaaaGcbaqcLbsacaWG4bGaey4kaSIaeqiUdeNaey4kaSIaaG OmaaaaaSqaaKqzGeGaamiEaiabg2da9iaaigdaaSqaaKqzGeGaam4A aaGaeyyeIuoaaaa@74B1@
The maximum likelihood estimate θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@ is the solution of the equation dlogL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGVbGaai4zaiaadYeaaOqaaKqzGeGa amizaiabeI7aXbaacqGH9aqpcaaIWaaaaa@40AF@ and is given by the solution of the following non-linear equation
2n θ n( 2θ+3 ) θ 2 +3θ+1 n x ¯ θ+1 + x=1 k f x x+θ+2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGOmaiaad6gaaOqaaKqzGeGaeqiUdehaaiabgkHiTKqb aoaalaaakeaajugibiaad6gajuaGdaqadaGcbaqcLbsacaaIYaGaeq iUdeNaey4kaSIaaG4maaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqC juaGdaahaaWcbeqaaKqzGeGaaGOmaaaacqGHRaWkcaaIZaGaeqiUde Naey4kaSIaaGymaaaacqGHsisljuaGdaWcaaGcbaqcLbsacaWGUbGa aGPaVlqadIhagaqeaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaaai abgUcaRKqbaoaaqahakeaajuaGdaWcaaGcbaqcLbsacaWGMbqcfa4a aSbaaSqaaKqzGeGaamiEaaWcbeaaaOqaaKqzGeGaamiEaiabgUcaRi abeI7aXjabgUcaRiaaikdaaaaaleaajugibiaadIhacqGH9aqpcaaI XaaaleaajugibiaadUgaaiabggHiLdGaeyypa0JaaGimaaaa@6C72@  (5.1.2)
where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b Gbaebaaaa@379A@ is the sample mean. This non-linear equation can be solved by any numerical iteration methods such as Newton- Raphson method, Bisection method, Regula –Falsi method etc. Ghitany et al. [20] showed that the MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ is consistent and asymptotically normal.

Method of Moment Estimate (MOME): Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiaacYcacaWG4bqc fa4aaSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaacYcacaaMc8Uaai Olaiaac6cacaGGUaGaaGPaVlaacYcacaWG4bqcfa4aaSbaaSqaaKqz GeGaamOBaaWcbeaaaaa@483C@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@  from the ZTPLD (2.2.1). Equating the first population moment about origin to the corresponding sample moment, the MOME θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  of ZTPLD (2.2.1) is the solution of the following cubic equation.
( x ¯ 1 ) θ 3 +( 3 x ¯ 4 ) θ 2 +( x ¯ 5 )θ2=0; x ¯ >1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaace WG4bGbaebacqGHsislcaaIXaaacaGLOaGaayzkaaGaeqiUde3aaWba aSqabeaacaaIZaaaaOGaey4kaSYaaeWaaeaacaaIZaGabmiEayaara GaeyOeI0IaaGinaaGaayjkaiaawMcaaiabeI7aXnaaCaaaleqabaGa aGOmaaaakiabgUcaRmaabmaabaGabmiEayaaraGaeyOeI0IaaGynaa GaayjkaiaawMcaaiabeI7aXjabgkHiTiaaikdacqGH9aqpcaaIWaGa aGPaVlaaykW7caGG7aGabmiEayaaraGaeyOpa4JaaGymaaaa@566F@ , where ( x ¯ 1 ) θ 3 +( 3 x ¯ 4 ) θ 2 +( x ¯ 5 )θ2=0; x ¯ >1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaace WG4bGbaebacqGHsislcaaIXaaacaGLOaGaayzkaaGaeqiUde3aaWba aSqabeaacaaIZaaaaOGaey4kaSYaaeWaaeaacaaIZaGabmiEayaara GaeyOeI0IaaGinaaGaayjkaiaawMcaaiabeI7aXnaaCaaaleqabaGa aGOmaaaakiabgUcaRmaabmaabaGabmiEayaaraGaeyOeI0IaaGynaa GaayjkaiaawMcaaiabeI7aXjabgkHiTiaaikdacqGH9aqpcaaIWaGa aGPaVlaaykW7caGG7aGabmiEayaaraGaeyOpa4JaaGymaaaa@566F@ is the sample mean. Ghitany et al. [20] showed that the MOME θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ is consistent and asymptotically normal.

Applications

In this section, both ZTPD and ZTPLD have been fitted to a number of data-sets using maximum likelihood estimates relating to demography, biological sciences, and social sciences to test their goodness of fits and it has been observed that in most of the cases ZTPLD gives much closer fits than ZTPD and in some cases ZTPD gives much closer fits than ZTPLD.

Mortality
Mortality does not depend only on biological factors; it depends upon the prevailing health conditions, medical facilities, the socio-economic and cultural factors. In developing and under-developed countries, the mortality among infants and children is found much higher than that among youngsters. The high infant mortality has thrown a serious challenge to the medical personnel and is considered as one of the sensitive position of existing medical and health facilities in the population.

Number of Neonatal Deaths

Observed Number of Mothers

Expected Frequency

ZTPD

ZTPLD

1

409

399.7

408.1

2

88

102.3

89.4

3
4
5

19
5
1

17.5 2.2 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaI3aGaaiOlaiaaiwdaaeaacaaIYaGaaiOlaiaaikda aeaacaaIWaGaaiOlaiaaiodaaaGaayzFaaaaaa@3E5B@

19.3
4.1 1.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGymaaaacaGL 9baaaaa@3B6D@

Total

522

522.0

522.0

ML Estimate

 

θ ^ =0.512047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaigdacaaI YaGaaGimaiaaisdacaaI3aaaaa@3EBD@

θ ^ =4.199697 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaisdacaGGUaGaaGymaiaaiMdacaaI 5aGaaGOnaiaaiMdacaaI3aaaaa@3ED7@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

3.464

0.145

d.f.

 

1

2

P-value

 

0.0627

0.9301

Table 1: The number of mothers of the rural area having at least one live birth and one neonatal death.

Number of Neonatal Deaths

Observed Number of Mothers

Expected Frequency

ZTPD

ZTPLD

1

71

66.5

72.3

2

32

35.1

28.4

3
4
5

7
5
3

12.3 3.3 0.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIYaGaaiOlaiaaiodaaeaacaaIZaGaaiOlaiaaioda aeaacaaIWaGaaiOlaiaaiIdaaaGaayzFaaaaaa@3E5B@

10.9
4.1 2.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGymaaqaaiaaikdacaGGUaGaaG4maaaacaGL 9baaaaa@3B70@

Total

118

118.0

118.0

ML Estimate

 

θ ^ =1.055102 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGimaiaaiwdacaaI 1aGaaGymaiaaicdacaaIYaaaaa@3EB8@

θ ^ =2.049609 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGimaiaaisdacaaI 5aGaaGOnaiaaicdacaaI5aaaaa@3EC8@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

0.696

2.274

d.f.

 

1

2

P-value

 

0.4041

0.3208

Table 2: The number of mothers of the estate area having at least one live birth and one neonatal death.

Number of Infant and Child Deaths

Observed Number of Mothers

Expected Frequency

ZTPD

ZTPLD

1

176

164.3

171.6

2

44

61.2

51.3

3
4
5

16
6
2

15.2 2.8 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaI1aGaaiOlaiaaikdaaeaacaaIYaGaaiOlaiaaiIda aeaacaaIWaGaaiOlaiaaiwdaaaGaayzFaaaaaa@3E5E@

15.0
4.3 1.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaG4maaqaaiaaigdacaGGUaGaaG4naaaacaGL 9baaaaa@3B75@

Total

244

244.0

244.0

ML Estimate

 

θ ^ =0.744522 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaisdacaaI 0aGaaGynaiaaikdacaaIYaaaaa@3EC2@

θ ^ =2.209411 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOmaiaaicdacaaI 5aGaaGinaiaaigdacaaIXaaaaa@3EBD@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

7.301

1.882

d.f.

 

1

2

P-value

 

0.0069

0.3902

Table 3: The number of mothers of the urban area with at least two live births by the number of infant and child deaths.

Number of Infant and Child Deaths

Observed Number of Mothers

Expected Frequency

ZTPD

ZTPLD

1

745

708.9

738.1

2

212

255.1

214.8

3

50

61.2

61.3

4
5
6

21
7
3

11.0 1.6 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIXaGaaiOlaiaaicdaaeaacaaIXaGaaiOlaiaaiAda aeaacaaIWaGaaiOlaiaaikdaaaGaayzFaaaaaa@3E52@

17.2
4.8 1.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaGioaaaacaGL 9baaaaa@3B7B@

Total

1038

1038.0

1038.0

ML Estimate

 

θ ^ =0.719783 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaigdacaaI 5aGaaG4naiaaiIdacaaIZaaaaa@3ECD@

θ ^ =3.007722 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaiodacaGGUaGaaGimaiaaicdacaaI 3aGaaG4naiaaikdacaaIYaaaaa@3EBF@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

37.046

4.773

d.f.

 

2

3

P-value

 

0.0

0.1892

Table 4: The number of mothers of the rural area with at least two live births by the number of infant and child deaths.

Number of Infant Deaths

Observed Number of Mothers

Expected Frequency

ZTPD

ZTPLD

1

683

659.0

674.4

2

145

177.4

154.1

3
4
5

29
11
5

31.8 4.3 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiodacaaIXaGaaiOlaiaaiIdaaeaacaaI0aGaaiOlaiaaioda aeaacaaIWaGaaiOlaiaaiwdaaaGaayzFaaaaaa@3E5F@

34.6
7.7 2.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiEdacaGGUaGaaG4naaqaaiaaikdacaGGUaGaaGOmaaaacaGL 9baaaaa@3B78@

Total

873

873.0

873.0

ML Estimate

 

θ ^ =0.538402 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaiodacaaI 4aGaaGinaiaaicdacaaIYaaaaa@3EC0@

θ ^ =4.00231 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaisdacaGGUaGaaGimaiaaicdacaaI YaGaaG4maiaaigdaaaa@3DFA@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

8.718

5.310

d.f.

 

1

2

P-value

 

0.0031

0.0703

Table 5: The number of literate mothers with at least one live birth by the number of infant deaths.

Number of Child Deaths

Observed Number of Mothers

Expected Frequency

ZTPD

ZTPLD

1

89

76.8

83.4

2

25

39.9

32.3

3
4
5
6

11
6
3
1

13.8 3.6 0.7 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIZaGaaiOlaiaaiIdaaeaacaaIZaGaaiOlaiaaiAda aeaacaaIWaGaaiOlaiaaiEdaaeaacaaIWaGaaiOlaiaaikdaaaGaay zFaaaaaa@408C@

12.2
4.5 1.6 0.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGynaaqaaiaaigdacaGGUaGaaGOnaaqaaiaa icdacaGGUaGaaGyoaaaacaGL9baaaaa@3DA6@

Total

135

135.0

135.0

ML Estimate

 

θ ^ =1.038289 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGimaiaaiodacaaI 4aGaaGOmaiaaiIdacaaI5aaaaa@3EC9@

θ ^ =2.089084 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGimaiaaiIdacaaI 5aGaaGimaiaaiIdacaaI0aaaaa@3EC9@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

7.90

3.428

d.f.

 

1

2

P-value

 

0.0049

0.1801

Table 6: The number of mothers of the completed fertility having experienced at least one child death.

Number Of Neonatal Deaths

Observed Number Of Mothers

Expected Frequency

ZTPD

ZTPLD

1

567

545.8

561.4

2

135

162.5

139.7

3

28

32.3

34.2

4
5

11
5

4.8 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaGOnaaaacaGL 9baaaaa@3B78@

8.2 2.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiIdacaGGUaGaaGOmaaqaaiaaikdacaGGUaGaaGOnaaaacaGL 9baaaaa@3B78@

Total

746

746.0

746.0

ML Estimate

 

θ ^ =0.595415 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaiMdacaaI 1aGaaGinaiaaigdacaaI1aaaaa@3EC7@

θ ^ =3.625737 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaiodacaGGUaGaaGOnaiaaikdacaaI 1aGaaG4naiaaiodacaaI3aaaaa@3ECB@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

26.855

3.839

d.f.

 

2

2

P-value

 

0.0

0.1467

Table 7: The number of mothers having at least one neonatal death.

In this section, an attempt has been made to test the suitability of ZTPD and ZTPLD in describing the neonatal deaths as well as of infant and child deaths experienced by mothers. The data-sets considered here are the data of Sri Lanka and India. The data-sets of Meegama et al. [21] have been used as the data of Sri Lanka whereas the data from the survey conducted by Lal [22] and the survey of Kadam Kuan, Patna, conducted in 1975 and referred to by Mishra [23] have been used as the data of India. It is obvious from the fittings of ZTPD and the ZTPLD that ZTPLD gives much closer fits in almost all cases except in Table 2. Hence, in case of demographic data, ZTPLD is a better alternative than ZTPD to model count data.

Biological Sciences
In this section, an attempt has been made to test the goodness of fit of both ZTPD and ZTPLD on many data- sets relating to biological sciences. It has been observed that ZTPLD gives much closer fits than ZTPD in almost all cases except in Table 11 regarding the distribution of the number of leaf spot grade of Ichinose variety of Mulberry. Thus in biological sciences ZTPLD is a better model than ZTPD to model zero-truncated count data.

Number of European Red Mites

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

38

28.7

36.1

2

17

25.7

20.5

3

10

15.3

11.2

4
5
6
7
8

9
3
2
1
0

6.9 2.5 0.7 0.2 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiAdacaGGUaGaaGyoaaqaaiaaikdacaGGUaGaaGynaaqaaiaa icdacaGGUaGaaG4naaqaaiaaicdacaGGUaGaaGOmaaqaaiaaicdaca GGUaGaaGymaaaacaGL9baaaaa@41FB@

5.9
3.1 1.6 0.8 0.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiodacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGOnaaqaaiaa icdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaGioaaaacaGL9baaaa a@3FCF@

Total

80

80.0

80.0

ML Estimate

 

θ ^ =1.791615 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaG4naiaaiMdacaaI XaGaaGOnaiaaigdacaaI1aaaaa@3EC8@

θ ^ =1.185582 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGymaiaaiIdacaaI 1aGaaGynaiaaiIdacaaIYaaaaa@3EC8@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

9.827

2.467

d.f.

 

2

3

P-value

 

0.0073

0.4813

Table 8: Number of European red mites on apple leaves, reported by Garman [24].

Number of Yeast Cells Counts Per Mm Square

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

128

121.3

127.6

2

37

49.2

40.9

3
4
5
6

18
3
1
0

13.3 2.7 0.4 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIZaGaaiOlaiaaiodaaeaacaaIYaGaaiOlaiaaiEda aeaacaaIWaGaaiOlaiaaisdaaeaacaaIWaGaaiOlaiaaigdaaaGaay zFaaaaaa@4083@

12.8 4.0 1.2 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIYaGaaiOlaiaaiIdaaeaacaaI0aGaaiOlaiaaicda aeaacaaIXaGaaiOlaiaaikdaaeaacaaIWaGaaiOlaiaaiwdaaaGaay zFaaaaaa@4085@

Total

187

187.0

187.0

ML Estimate

 

θ ^ =0.811276 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGioaiaaigdacaaI XaGaaGOmaiaaiEdacaaI2aaaaa@3EC3@

θ ^ =2.667323 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaiAdacaaI 3aGaaG4maiaaikdacaaIZaaaaa@3EC7@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

5.228

1.034

d.f.

 

1

1

P-value

 

0.0222

0.3092

Table 9: Number of yeast cell counts observed per mm square, reported by Student [25].

Number of Fly Eggs

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

22

15.3

26.8

2

18

21.9

19.8

3

18

20.8

13.9

4

11

14.9

9.5

5

9

8.5

6.4

6
7
8
9

6
3
0
1

4.1 1.7 0.6 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaG4naaqaaiaa icdacaGGUaGaaGOnaaqaaiaaicdacaGGUaGaaG4maaaacaGL9baaaa a@3FCA@

4.2 2.7 1.7 3.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGOmaaqaaiaaikdacaGGUaGaaG4naaqaaiaa igdacaGGUaGaaG4naaqaaiaaiodacaGGUaGaaGimaaaacaGL9baaaa a@3FCE@

Total

88

88.0

88.0

ML Estimate

 

θ ^ =2.860402 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGioaiaaiAdacaaI WaGaaGinaiaaicdacaaIYaaaaa@3EC0@

θ ^ =0.718559 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaigdacaaI 4aGaaGynaiaaiwdacaaI5aaaaa@3ECD@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

6.677

3.743

d.f.

 

4

4

P-value

 

0.1540

0.4419

Table 10: The number of counts of flower heads as per the number of fly eggs reported by Finney & Varley [26].

Number Of Leaf Spot Grade

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

18

14.2

23.0

2

15

18.7

16.3

3

10

16.5

11.1

4

14

10.9

7.3

5

13

9.7

12.4

Total

70

70.0

70.0

ML Estimate

 

θ ^ =2.639984 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaiodacaaI 5aGaaGyoaiaaiIdacaaI0aaaaa@3ED3@

θ ^ =0.781902 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaiIdacaaI XaGaaGyoaiaaicdacaaIYaaaaa@3EC5@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

6.311

7.476

d.f.

 

3

3

P-value

 

0.0974

0.0582

Table 11: The number of leaf spot grade of Ichinose variety of Mulberry, reported by Khurshid [27].

Number of Leaf Spot Grade

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

37

28.5

36.7

2

16

26.7

21.4

3

15

16.7

12.0

4
5

8
8

7.8 4.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiEdacaGGUaGaaGioaaqaaiaaisdacaGGUaGaaGOmaaaacaGL 9baaaaa@3B7B@

6.6
7.3

Total

84

84.0

84.0

ML Estimate

 

θ ^ =1.874567 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGioaiaaiEdacaaI 0aGaaGynaiaaiAdacaaI3aaaaa@3ED0@

θ ^ =1.130211 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGymaiaaiodacaaI WaGaaGOmaiaaigdacaaIXaaaaa@3EB3@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

8.329

2.477

d.f.

 

2

3

P-value

 

0.0155

0.4795

Table 12: The number of leaf spot grade of Kokuso-20 variety of Mulberry, reported by Khurshid [27].

Number of Sites With Particles

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

122

115.9

124.8

2

50

57.4

46.8

3

18

18.9

17.1

4
5

4
4

4.7 1.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaG4naaqaaiaaigdacaGGUaGaaGymaaaacaGL 9baaaaa@3B73@

6.1 3.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiAdacaGGUaGaaGymaaqaaiaaiodacaGGUaGaaGOmaaaacaGL 9baaaaa@3B72@

Total

198

198.0

198.0

ML Estimate

 

θ ^ =0.990586 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGyoaiaaiMdacaaI WaGaaGynaiaaiIdacaaI2aaaaa@3ECF@

θ ^ =2.18307 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGymaiaaiIdacaaI ZaGaaGimaiaaiEdaaaa@3E05@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

2.14

0.51

d.f.

 

2

2

P-value

 

0.3430

0.7749

Table 13: The number of counts of sites with particles from Immuno gold data, reported by Mathews [28].

Number of Times Hares Caught

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

184

176.6

182.6

2

55

66.0

55.3

3
4
5

14
4
4

16.6 3.1 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaI2aGaaiOlaiaaiAdaaeaacaaIZaGaaiOlaiaaigda aeaacaaIWaGaaiOlaiaaiEdaaaGaayzFaaaaaa@3E5F@

16.4
4.8 1.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaisdacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaGyoaaaacaGL 9baaaaa@3B7C@

Total

261

261.0

261.0

ML Estimate

 

θ ^ =0.756171 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaiwdacaaI 2aGaaGymaiaaiEdacaaIXaaaaa@3EC5@

θ ^ =2.863957 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGioaiaaiAdacaaI ZaGaaGyoaiaaiwdacaaI3aaaaa@3ED2@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

2.45

0.61

d.f.

 

1

2

P-value

 

0.1175

0.7371

Table 14: The number of snowshoe hares counts captured over 7 days, reported by Keith & Meslow [29].

Social Sciences
In this section, an attempt has been made to test the goodness of fit test of both ZTPD and ZTPLD on many data-sets relating to social sciences, such as migration, number of accidents and free-forming small group size. It has been observed that the ZTPD gives much closer fits than ZTPLD in almost all cases except the distribution of the number of household having at least one migrant in Table 15.

X

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

375

354.0

379.0

2

143

167.7

137.2

3

49

53.0

48.4

4
5
6
7
8

17
2
2
1
1

12.5 2.4 0.4 0.1 0.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIYaGaaiOlaiaaiwdaaeaacaaIYaGaaiOlaiaaisda aeaacaaIWaGaaiOlaiaaisdaaeaacaaIWaGaaiOlaiaaigdaaeaaca aIWaGaaiOlaiaaicdaaaGaayzFaaaaaa@42A8@

16.8
5.7 1.9 0.6 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiwdacaGGUaGaaG4naaqaaiaaigdacaGGUaGaaGyoaaqaaiaa icdacaGGUaGaaGOnaaqaaiaaicdacaGGUaGaaG4maaaacaGL9baaaa a@3FD3@

Total

590

590.0

590.0

ML Estimate

 

θ ^ =0.947486 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGyoaiaaisdacaaI 3aGaaGinaiaaiIdacaaI2aaaaa@3ED0@

θ ^ =2.284782 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOmaiaaiIdacaaI 0aGaaG4naiaaiIdacaaIYaaaaa@3ECB@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

8.933

1.031

d.f.

 

2

3

P-value

 

0.0115

0.7937

Table 15: Number of households having at least one migrant according to the number of migrants, reported by Sing & Yadav [30].

Number of Accidents

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

2039

2034.2

2050.4

2

312

319.5

291.7

3
4
5

35
3
1

33.5
2.6
0.2

41.1
5.8
1.0

Total

2390

2390.0

2390.0

ML Estimate

 

θ ^ =0.314125 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4maiaaigdacaaI 0aGaaGymaiaaikdacaaI1aaaaa@3EBA@

θ ^ =6.749732 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaiAdacaGGUaGaaG4naiaaisdacaaI 5aGaaG4naiaaiodacaaIYaaaaa@3ED0@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

0.387

3.128

d.f.

 

1

1

P-value

 

0.5339

0.0769

Table 16: Number of workers according to the number of accidents, reported by Mir & Ahmad [31].

Number of Pairs of Running Shoes

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

18

17.7

24.1

2

18

18.5

15.0

3

12

12.9

9.0

4
5

7
5

6.7 4.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiAdacaGGUaGaaG4naaqaaiaaisdacaGGUaGaaGOmaaaacaGL 9baaaaa@3B79@

5.2
6.2

Total

60

60.0

60.0

ML Estimate

 

θ ^ =2.087937 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGimaiaaiIdacaaI 3aGaaGyoaiaaiodacaaI3aaaaa@3ECE@

θ ^ =1.004473 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGimaiaaicdacaaI 0aGaaGinaiaaiEdacaaIZaaaaa@3EBD@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

0.191

3.998

d.f.

 

2

3

P-value

 

0.9089

0.2617

Table 17: Number of counts of pairs of running shoes owned by 60 members of an athletics club, reported by Simonoff [32].

Group Size

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

1486

1500.5

1592.8

2

694

669.6

551.8

3

195

199.2

186.5

4

37

44.4

61.9

5
6

10
1

7.9 1.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiEdacaGGUaGaaGyoaaqaaiaaigdacaGGUaGaaG4maaaacaGL 9baaaaa@3B7A@

20.3 9.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaikdacaaIWaGaaiOlaiaaiodaaeaacaaI5aGaaiOlaiaaiAda aaGaayzFaaaaaa@3C34@

Total

2423

2423.0

2423.0

ML Estimate

 

θ ^ =0.892496 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGioaiaaiMdacaaI YaGaaGinaiaaiMdacaaI2aaaaa@3ED0@

θ ^ =2.419103 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGinaiaaigdacaaI 5aGaaGymaiaaicdacaaIZaaaaa@3EBE@

 

2.702

66.155

d.f.

 

3

3

P-value

 

0.4399

0.0

Table 18: Number of free forming small group size according to the group size, reported by Coleman & James [33].

Group Size

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

316

316.4

335.8

2

141

140.7

116.0

3

44

41.7

39.1

4
5

5
4

9.3 1.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiMdacaGGUaGaaG4maaqaaiaaigdacaGGUaGaaGyoaaaacaGL 9baaaaa@3B7C@

12.9 6.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIYaGaaiOlaiaaiMdaaeaacaaI2aGaaiOlaiaaikda aaGaayzFaaaaaa@3C34@

Total

510

510.0

510.0

ML Estimate

 

θ ^ =0.889458 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGioaiaaiIdacaaI 5aGaaGinaiaaiwdacaaI4aaaaa@3ED4@

θ ^ =2.428125 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGinaiaaikdacaaI 4aGaaGymaiaaikdacaaI1aaaaa@3EC2@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

0.558

12.481

d.f.

 

2

2

P-value

 

0.7565

0.0019

Table 19: Number of free forming small group size according to the group size, reported by Coleman & James [33].

Group Size

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

306

302.5

322.5

2

132

139.5

114.6

3

47

42.9

39.7

4
5

10
2

9.9 2.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaiMdacaGGUaGaaGyoaaqaaiaaikdacaGGUaGaaGymaaaacaGL 9baaaaa@3B7B@

13.5 6.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIZaGaaiOlaiaaiwdaaeaacaaI2aGaaiOlaiaaiIda aaGaayzFaaaaaa@3C37@

Total

497

497.0

497.0

ML Estimate

 

θ ^ =0.922509 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGyoaiaaikdacaaI YaGaaGynaiaaicdacaaI5aaaaa@3EC5@

θ ^ =2.341269 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaG4maiaaisdacaaI XaGaaGOmaiaaiAdacaaI5aaaaa@3EC5@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

0.834

8.220

d.f.

 

2

2

P-value

 

0.6590

0.0164

Table 20: Number of free forming small group size according to the group size, reported by Coleman & James [33].

Group Size

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

1

305

307.2

327.7

2

144

142.9

117.3

3

50

44.3

40.9

4
5
6

5
2
1

10.3 1.9 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaIWaGaaiOlaiaaiodaaeaacaaIXaGaaiOlaiaaiMda aeaacaaIWaGaaiOlaiaaiodaaaGaayzFaaaaaa@3E58@

14.0 4.7 2.4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaaigdacaaI0aGaaiOlaiaaicdaaeaacaaI0aGaaiOlaiaaiEda aeaacaaIYaGaaiOlaiaaisdaaaGaayzFaaaaaa@3E5D@

Total

507

507.0

507.0

ML Estimate

 

θ ^ =0.930664 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGyoaiaaiodacaaI WaGaaGOnaiaaiAdacaaI0aaaaa@3EC6@

θ ^ =2.31943 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaG4maiaaigdacaaI 5aGaaGinaiaaiodaaaa@3E06@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaaaaaaa@3896@

 

2.376

17.806

d.f.

 

2

2

P-value

 

0.3048

0.0001

Table 21: Number of free forming small group size according to the group size, reported by Coleman & James [33].

Conclusion

In this paper, the nature and behavior of ZTPD and ZTPLD have been studied by drawing different graphs for the different values of its parameter. A general expression for the th factorial moment has been given and the first four moments about origin has been obtained. Also a very simple and easy method for finding moments of ZTPLD has been suggested. An attempt has been made to study the goodness of fit of both ZTPD and ZTPLD to count data relating to demography, biological sciences, and social sciences and it has been found that ZTPLD is a better model than the ZTPD in almost all data-sets relating to mortality and biological sciences whereas ZTPD is a better model than ZTPLD in almost all data-sets relating to social sciences.. Thus, ZTPLD has an advantage over ZTPD for modeling zero-truncated count data in mortality and biological sciences whereas ZTPD has an advantage over ZTPLD for modeling zer-truncated count data in social sciences.

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