ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 5 - 2016
On Zero-Truncation of Poisson, Poisson-Lindley and Poisson-Sujatha Distributions and their Applications
Rama Shanker1* and Hagos Fesshaye2
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
Received:March 17, 2016 | Published: April 15, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H (2016) On Zero-Truncation of Poisson, Poisson-Lindley and Poisson-Sujatha Distributions and their Applications. Biom Biostat Int J 3(5): 00072. DOI:10.15406/bbij.2016.03.00072

Abstract

In the present paper, firstly the nature of zero-truncated Poisson distribution (ZTPD), zero-truncated Poisson-Lindley distribution (ZTPLD) and zero-truncated Poisson-Sujatha distribution (ZTPSD) have been discussed with graphs of their probability functions for different values of their parameter. Over-dispersion, equi-dispersion and under-dispersion of ZTPLD and ZTPSD have been discussed using index of dispersion. A simple and interesting method of finding moments of ZTPSD has been suggested and thus the first two moments about origin and the variance have been obtained. The estimation of parameter of ZTPD, ZTPLD and ZTPSD has been discussed using maximum likelihood estimation and method of moments.

The goodness of fit of ZTPD, ZTPLD, and ZTPSD using maximum likelihood estimate in zero-truncated data arising from demography, biological sciences and migration has been discussed and observed that in most data-sets relating to demography and biological sciences, ZTPSD gives better fit than ZTPD and ZTPLD.

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

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaaaa@3DEB@  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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaaaa@3DEB@  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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiaadcfadaWgaaqcfasaaiaaicdaaKqbagqaamaabmaaba GaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaaaeaacaaIXaGaeyOe I0IaamiuamaaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacaaIWa Gaai4oaiabeI7aXbGaayjkaiaawMcaaaaacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaacUdacaWG4bGaeyypa0JaaGymaiaacYcacaaIYa GaaiilaiaaiodacaGGSaGaaiOlaiaac6cacaGGUaaaaa@5E70@                                                        (1.1)

The Poisson-Lindley distribution (PLD) 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGdaqadaqaaiaadIhacqGHRaWkcqaH4oqC cqGHRaWkcaaIYaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4oqCcq GHRaWkcaaIXaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamiEaiab gUcaRiaaiodaaaaaaKqbakaaykW7caaMc8UaaGPaVlaaykW7caGG7a GaamiEaiabg2da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiil aiaaiodacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaaykW7caaMc8 UaeqiUdeNaeyOpa4JaaGimaaaa@6953@                                  (1.2)

has been  introduced by Sankaran [1] to model count data. Recently, Shanker & Hagos [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 PLD 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 [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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aabmaabaGaeq4UdWMaai4oaiabeI7aXbGaayjkaiaawMcaaiabg2da 9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaKqbagaacq aH4oqCcqGHRaWkcaaIXaaaamaabmaabaGaaGymaiabgUcaRiabeU7a SbGaayjkaiaawMcaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4o qCcqaH7oaBaaqcfaOaaGPaVlaacUdacaaMc8UaaGPaVlaaykW7cqaH 7oaBcqGH+aGpcaaIWaGaaGPaVlaaykW7caGGSaGaeqiUdeNaeyOpa4 JaaGimaaaa@6173@                                (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 for modeling some lifetime data. Recently, Shanker et al. [5] showed that (1.3) is not always a better model than the exponential distribution for modeling lifetime’s data. In fact, Shanker et al. [6] has done a very extensive and comparative study on modeling of lifetime data using exponential and Lindley distributions and discussed various lifetime data-sets 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 [7] and its various properties have been discussed by them.  The Lindley distribution and the PLD have been generalized by many researchers.  Shanker & Mishra [8] obtained a two parameter Poisson-Lindley distribution by compounding Poisson distribution with a two parameter Lindley distribution introduced by Shanker & Mishra [9]. A quasi Poisson-Lindley distribution has been introduced by Shanker & Mishra [10] by compounding Poisson distribution with a quasi Lindley distribution introduced by Shanker & Mishra [11]. Shanker et al. [12] 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. [13]. Further, Shanker & Tekie [14] obtained a new quasi Poisson-Lindley distribution by compounding Poisson distribution with a new quasi Lindley distribution introduced by Shanker & Amanuel [15].

Recently Shanker [16] has obtained Poisson-Sujatha distribution (PSD) having p.m.f. 

P 0 ( x;θ )= θ 3 θ 2 +θ+2 x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ( θ+1 ) x+3 ;x=0,1,2,3,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaikdaaaWaaSaaaeaaca WG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqadaqaaiab eI7aXjabgUcaRiaaisdaaiaawIcacaGLPaaacaWG4bGaey4kaSYaae WaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa iodacqaH4oqCcqGHRaWkcaaI0aaacaGLOaGaayzkaaaabaWaaeWaae aacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaWbaaeqajuai baGaamiEaiabgUcaRiaaiodaaaaaaKqbakaaykW7caaMc8UaaGPaVl aaykW7caGG7aGaamiEaiabg2da9iaaicdacaGGSaGaaGymaiaacYca caaIYaGaaiilaiaaiodacaGGSaGaaiOlaiaac6cacaGGUaGaaiilai aaykW7caaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@7E33@       (1.4)

to model count data in different fields of knowledge. 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 Shanker R [17] with probability density function (p.d.f.)

g( λ;θ )= θ 3 θ 2 +θ+2 ( 1+λ+ λ 2 ) e θλ ;λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aabmaabaGaeq4UdWMaai4oaiabeI7aXbGaayjkaiaawMcaaiabg2da 9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaKqbagaacq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiabeI7aXjab gUcaRiaaikdaaaWaaeWaaeaacaaIXaGaey4kaSIaeq4UdWMaey4kaS Iaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaa caWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlabeU7aSb aajuaGcaaMc8UaaGPaVlaaykW7caaMc8Uaai4oaiaaykW7cqaH7oaB cqGH+aGpcaaIWaGaaGPaVlaaykW7caGGSaGaeqiUdeNaeyOpa4JaaG imaaaa@6CED@                      (1.5)

Detailed discussion about its various properties, estimation of the parameter and applications for modeling lifetime data has been mentioned in Shanker [17] and shown by Shanker [17] that (1.5) is a better model than the exponential and Lindley [3] distributions for modeling lifetime data.  Shanker & Hagos [18,19] has also obtained the size-biased and zero-truncated version of PSD and discussed their properties, estimation of parameter and applications in different fields of knowledge.   

In this paper, the nature of zero-truncated Poisson distribution (ZTPD), zero-truncated Poisson-Lindley distribution (ZTPLD) and zero-truncated Poisson-Sujatha distribution has been compared and studied using graphs for different values of their parameter. A simple method for obtaining the moments of ZTPSD has been suggested and the first two moments about origin and variance have been obtained. ZTPD, ZTPLD and ZTPSD have been fitted to a number of data -sets from demography and biological sciences to study their goodness of fit and superiority of one over the others.

Zero-Truncated Poisson, Poisson-Lindley and Poisson-Sujatha Distributions

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaamiEaaaaaKqbagaadaqadaqaaiaadwgadaahaaqabKqb GeaacqaH4oqCaaqcfaOaeyOeI0IaaGymaaGaayjkaiaawMcaaiaayk W7caWG4bGaaiyiaaaacaaMc8UaaGPaVlaaykW7caGG7aGaamiEaiab g2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6caca GGUaGaaiOlaiaac6cacaGGSaGaaGPaVlabeI7aXjabg6da+iaaicda aaa@6091@                                    (2.1.1)

was obtained independently by Plackett [20] and David & Johnson [21] to model count data excluding 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 [22,23]. Tate & Goen [24] have 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaIXaaaamaala aabaGaamiEaiabgUcaRiabeI7aXjabgUcaRiaaikdaaeaadaqadaqa aiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaadaahaaqabKqbGe aacaWG4baaaaaajuaGcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa cUdacaWG4bGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodaca GGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcacaaMc8UaeqiUdeNa eyOpa4JaaGimaaaa@6DC7@              (2.2.1)

was obtained by Ghitany et al. [25] to model count data for the missing zeros.

Shanker et al. [2] have done extensive study on the comparison of ZTPD and ZTPLD with respect to their applications in data - sets excluding zero-counts and showed that in demography and biological sciences ZTPLD gives better fit than ZTPD while in social sciences ZTPD gives better fit than ZTPLD.

Zero-Truncated Poisson-Sujatha Distribution (ZTPSD)

Using (1.1) and (1.4), the p.m.f. of zero-truncated Poisson-Sujatha distribution (ZTPSD) can be obtained as

P 3 ( x;θ )= θ 3 θ 4 +4 θ 3 +10 θ 2 +7θ+2 x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ( θ+1 ) x ;x=1,2,3,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI 0aaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIZa aaaKqbakabgUcaRiaaigdacaaIWaGaeqiUde3aaWbaaeqajuaibaGa aGOmaaaajuaGcqGHRaWkcaaI3aGaeqiUdeNaey4kaSIaaGOmaaaada WcaaqaaiaadIhadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaa bmaabaGaeqiUdeNaey4kaSIaaGinaaGaayjkaiaawMcaaiaadIhacq GHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOa ey4kaSIaaG4maiabeI7aXjabgUcaRiaaisdaaiaawIcacaGLPaaaae aadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaadaah aaqabKqbGeaacaWG4baaaaaajuaGcaaMc8UaaGPaVlaaykW7caaMc8 Uaai4oaiaadIhacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaG4m aiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaaGPaVlaaykW7cqaH4o qCcqGH+aGpcaaIWaaaaa@8688@       (2.3.1)

The ZTPSD can also arise from the size-biased Poisson distribution (SBPD) with p.m.f.

g( 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aabmaabaGaamiEaiaacYhacqaH7oaBaiaawIcacaGLPaaacqGH9aqp daWcaaqaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH7oaBaaqcfa Oaeq4UdW2aaWbaaeqajuaibaGaamiEaiabgkHiTiaaigdaaaaajuaG baWaaeWaaeaacaWG4bGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaacg caaaGaaGPaVlaaykW7caaMc8UaaGPaVlaacUdacaWG4bGaeyypa0Ja aGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOlaiaac6caca GGUaGaaGPaVlaaykW7caGGSaGaeq4UdWMaeyOpa4JaaGimaaaa@62B7@                                              (2.3.2)

When its parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@37AA@ follows a distribution having p.d.f.

h( λ;θ )= θ 3 θ 4 +4 θ 3 +10 θ 2 +7θ+2 [ ( θ+1 ) 2 λ 2 +( θ+1 )( θ+3 )λ+( θ 2 +3θ+4 ) ] e θλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaeq4UdWMaai4oaiabeI7aXbGaayjkaiaawMcaaiabg2da 9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaaaKqbagaacq aH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakabgUcaRiaaisdacqaH 4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaigdacaaIWa GaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI3aGa eqiUdeNaey4kaSIaaGOmaaaadaWadaqaamaabmaabaGaeqiUdeNaey 4kaSIaaGymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaqc faOaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqada qaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaadaqadaqaaiab eI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaacqaH7oaBcqGHRaWkda qadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIa aG4maiabeI7aXjabgUcaRiaaisdaaiaawIcacaGLPaaaaiaawUfaca GLDbaacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlab eU7aSbaaaaa@7E3D@ ,λ>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiilai abeU7aSjabg6da+iaaicdacaGGSaGaaGPaVlabeI7aXjabg6da+iaa icdaaaa@405D@     (2.3.3)

The p.m.f. of ZTPSD is thus can be obtained as

P( x;θ )= 0 g( x|λ ) h( λ;θ )dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aabmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaacqGH9aqp daWdXbqaaiaadEgadaqadaqaaiaadIhacaGG8bGaeq4UdWgacaGLOa GaayzkaaaajuaibaGaaGimaaqaaiabg6HiLcqcfaOaey4kIipacqGH flY1caWGObWaaeWaaeaacqaH7oaBcaGG7aGaeqiUdehacaGLOaGaay zkaaGaamizaiabeU7aSbaa@543E@              = 0 e λ λ x1 ( x1 )! θ 3 θ 4 +4 θ 3 +10 θ 2 +7θ+2 [ ( θ+1 ) 2 λ 2 +( θ+1 )( θ+3 )λ+( θ 2 +3θ+4 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 Zaa8qCaeaadaWcaaqaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH 7oaBaaqcfaOaeq4UdW2aaWbaaeqajuaibaGaamiEaiabgkHiTiaaig daaaaajuaGbaWaaeWaaeaacaWG4bGaeyOeI0IaaGymaaGaayjkaiaa wMcaaiaacgcaaaaajuaibaGaaGimaaqaaiabg6HiLcqcfaOaey4kIi pacqGHflY1daWcaaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaaa juaGbaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkca aI0aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI XaGaaGimaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaS IaaG4naiabeI7aXjabgUcaRiaaikdaaaWaamWaaeaadaqadaqaaiab eI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaadaahaaqabKqbGeaaca aIYaaaaKqbakabeU7aSnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4k aSYaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaWaae WaaeaacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaGaeq4UdWMa ey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbak abgUcaRiaaiodacqaH4oqCcqGHRaWkcaaI0aaacaGLOaGaayzkaaaa caGLBbGaayzxaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXj aaykW7cqaH7oaBaaqcfaOaamizaiabeU7aSbaa@90CE@  (2.3.4)                                                                                                                       

= θ 3 ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 )( x1 )! 0 e ( θ+1 )λ [ ( θ+1 ) 2 λ x+21 +( θ+1 )( θ+3 ) λ x+11 +( θ 2 +3θ+4 ) λ x1 ]dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaamaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkca aI0aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI XaGaaGimaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaS IaaG4naiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaadaqadaqa aiaadIhacqGHsislcaaIXaaacaGLOaGaayzkaaGaaiyiaaaadaWdXb qaaiaadwgadaahaaqcfasabeaacqGHsisljuaGdaqadaqcfasaaiab eI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaacqaH7oaBaaaabaGaaG imaaqaaiabg6HiLcqcfaOaey4kIipacqGHflY1daWadaabaeqabaWa aeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaWbaae qajuaibaGaaGOmaaaajuaGcqaH7oaBdaahaaqabKqbGeaacaWG4bGa ey4kaSIaaGOmaiabgkHiTiaaigdaaaqcfaOaey4kaSYaaeWaaeaacq aH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqC cqGHRaWkcaaIZaaacaGLOaGaayzkaaGaeq4UdW2aaWbaaeqajuaiba GaamiEaiabgUcaRiaaigdacqGHsislcaaIXaaaaaqcfayaaiabgUca RmaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRa WkcaaIZaGaeqiUdeNaey4kaSIaaGinaaGaayjkaiaawMcaaiabeU7a SnaaCaaabeqcfasaaiaadIhacqGHsislcaaIXaaaaaaajuaGcaGLBb GaayzxaaGaamizaiabeU7aSbaa@9827@

= θ 3 ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 ) [ ( x+1 )x ( θ+1 ) x + ( θ+3 )x ( θ+1 ) x + θ 2 +3θ+4 ( θ+1 ) x ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaamaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkca aI0aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI XaGaaGimaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaS IaaG4naiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaaaaWaamWa aeaadaWcaaqaamaabmaabaGaamiEaiabgUcaRiaaigdaaiaawIcaca GLPaaacaWG4baabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGL OaGaayzkaaWaaWbaaeqajuaibaGaamiEaaaaaaqcfaOaey4kaSYaaS aaaeaadaqadaqaaiabeI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaa caWG4baabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaay zkaaWaaWbaaeqajuaibaGaamiEaaaaaaqcfaOaey4kaSYaaSaaaeaa cqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiodacq aH4oqCcqGHRaWkcaaI0aaabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaI XaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamiEaaaaaaaajuaGca GLBbGaayzxaaaaaa@7AA0@

= θ 3 ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 ) x 2 +( θ+4 )x+( θ 2 +3θ+4 ) ( θ+1 ) x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaamaa bmaabaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkca aI0aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaI XaGaaGimaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaS IaaG4naiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaaaaWaaSaa aeaacaWG4bWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqada qaaiabeI7aXjabgUcaRiaaisdaaiaawIcacaGLPaaacaWG4bGaey4k aSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgU caRiaaiodacqaH4oqCcqGHRaWkcaaI0aaacaGLOaGaayzkaaaabaWa aeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaWaaWbaae qajuaibaGaamiEaaaaaaaaaa@6993@

x=1,2,3,...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6ca caGGUaGaaiOlaiaacYcacaaMc8UaaGPaVlabeI7aXjabg6da+iaaic daaaa@461F@

which is the p.m.f. of ZTPSD with parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Shanker & Hagos [19] have detailed study about its mathematical and statistical properties, estimation of parameter, and applications and showed that in many ways it has interesting advantage over ZTPLD and ZTPD.

To study the nature and behaviors of ZTPD, ZTPLD and ZTPSD for different values of their parameter, a number of graphs of their probability functions have been drawn and presented in Figure 1.

Moments and Related Measures of ZTPSD

The r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th moment about origin of ZTPSD (2.3.1) can be obtained as

μ r =E[ E( X r |λ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKazfa4=baGaamOCaaqcfayabaWaaWbaaeqajuaibaGamai4 gkdiIcaajuaGcqGH9aqpcaWGfbWaamWaaeaacaWGfbWaaeWaaeaaca WGybWaaWbaaeqajuaibaGaamOCaaaajuaGcaGG8bGaeq4UdWgacaGL OaGaayzkaaaacaGLBbGaayzxaaaaaa@4B12@       

= θ 3 θ 4 +4 θ 3 +10 θ 2 +7θ+2 0 [ x=1 x r e λ λ x1 ( x1 )! ] [ ( θ+1 ) 2 λ 2 +( θ+1 )( θ+3 )λ +( θ 2 +3θ+4 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaaqcfayaaiab eI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaGinaiabeI 7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGymaiaaicda cqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiEdacq aH4oqCcqGHRaWkcaaIYaaaamaapehabaWaamWaaeaadaaeWbqaaiaa dIhadaahaaqabeaacaWGYbaaaaqcfasaaiaadIhacqGH9aqpcaaIXa aabaGaeyOhIukajuaGcqGHris5amaalaaabaGaamyzamaaCaaabeqc fasaaiabgkHiTiabeU7aSbaajuaGcqaH7oaBdaahaaqabKqbGeaaca WG4bGaeyOeI0IaaGymaaaaaKqbagaadaqadaqaaiaadIhacqGHsisl caaIXaaacaGLOaGaayzkaaGaaiyiaaaaaiaawUfacaGLDbaaaKqbGe aacaaIWaaabaGaeyOhIukajuaGcqGHRiI8aiabgwSixpaadmaaeaqa beaadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaada ahaaqabKqbGeaacaaIYaaaaKqbakabeU7aSnaaCaaabeqcfasaaiaa ikdaaaqcfaOaey4kaSYaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaaca GLOaGaayzkaaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGa ayzkaaGaeq4UdWgabaGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabK qbGeaacaaIYaaaaKqbakabgUcaRiaaiodacqaH4oqCcqGHRaWkcaaI 0aaacaGLOaGaayzkaaaaaiaawUfacaGLDbaacaWGLbWaaWbaaeqaju aibaGaeyOeI0IaeqiUdeNaaGPaVlabeU7aSbaajuaGcaWGKbGaeq4U dWgaaa@9BF2@  (3.1)                                                                                   

Clearly the expression under the bracket in (3.1) is the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th moment about origin of the SBPD. Taking r=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iaaigdaaaa@38AE@  in (3.1) and using the first moment about origin of the SBPD, the first moment about origin of the ZTPSD (2.3.1) can be obtained as

μ 1 = θ 3 θ 4 +4 θ 3 +10 θ 2 +7θ+2 0 ( λ+1 ) [ ( θ+1 ) 2 λ 2 +( θ+1 )( θ+3 )λ +( θ 2 +3θ+4 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabKqbGeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG 4maaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakab gUcaRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgU caRiaaigdacaaIWaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaG cqGHRaWkcaaI3aGaeqiUdeNaey4kaSIaaGOmaaaadaWdXbqaamaabm aabaGaeq4UdWMaey4kaSIaaGymaaGaayjkaiaawMcaaaqcfasaaiaa icdaaeaacqGHEisPaKqbakabgUIiYdWaamWaaqaabeqaamaabmaaba GaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaaCaaabeqcfasa aiaaikdaaaqcfaOaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaajuaGcq GHRaWkdaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaa daqadaqaaiabeI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaacqaH7o aBaeaacqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSIaaG4maiabeI7aXjabgUcaRiaaisdaaiaawIcaca GLPaaaaaGaay5waiaaw2faaiaadwgadaahaaqabKqbGeaacqGHsisl cqaH4oqCcaaMc8Uaeq4UdWgaaKqbakaadsgacqaH7oaBaaa@8BF6@

= θ 5 +5 θ 4 +15 θ 3 +25 θ 2 +20θ+6 θ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI1aaaaKqbakabgUca RiaaiwdacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakabgUcaRi aaigdacaaI1aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGH RaWkcaaIYaGaaGynaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfa Oaey4kaSIaaGOmaiaaicdacqaH4oqCcqGHRaWkcaaI2aaabaGaeqiU de3aaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakabgU caRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUca RiaaigdacaaIWaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcq GHRaWkcaaI3aGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaaaa aaa@670A@                                                               (3.2)

Again taking r=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iaaikdaaaa@38AF@  in (3.1) and using the second moment about origin of the SBPD, the second moment about origin of the ZTPSD (2.3) can be obtained as

μ 2 = θ 3 θ 4 +4 θ 3 +10 θ 2 +7θ+2 0 ( λ 2 +3λ+1 ) [ ( θ+1 ) 2 λ 2 +( θ+1 )( θ+3 )λ +( θ 2 +3θ+4 ) ] e θλ dλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabKqbGeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaG 4maaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakab gUcaRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgU caRiaaigdacaaIWaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaG cqGHRaWkcaaI3aGaeqiUdeNaey4kaSIaaGOmaaaadaWdXbqaamaabm aabaGaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI ZaGaeq4UdWMaey4kaSIaaGymaaGaayjkaiaawMcaaaqcfasaaiaaic daaeaacqGHEisPaKqbakabgUIiYdWaamWaaqaabeqaamaabmaabaGa eqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaaCaaabeqcfasaai aaikdaaaqcfaOaeq4UdW2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkdaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaada qadaqaaiabeI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaacqaH7oaB aeaacqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaa qcfaOaey4kaSIaaG4maiabeI7aXjabgUcaRiaaisdaaiaawIcacaGL PaaaaaGaay5waiaaw2faaiaadwgadaahaaqabKqbGeaacqGHsislcq aH4oqCcaaMc8Uaeq4UdWgaaKqbakaadsgacqaH7oaBaaa@90E4@

= θ 6 +7 θ 5 +27 θ 4 +73 θ 3 +112 θ 2 +84θ+24 θ 2 ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI2aaaaKqbakabgUca RiaaiEdacqaH4oqCdaahaaqabKqbGeaacaaI1aaaaKqbakabgUcaRi aaikdacaaI3aGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGcqGH RaWkcaaI3aGaaG4maiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa Oaey4kaSIaaGymaiaaigdacaaIYaGaeqiUde3aaWbaaeqajuaibaGa aGOmaaaajuaGcqGHRaWkcaaI4aGaaGinaiabeI7aXjabgUcaRiaaik dacaaI0aaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqa daqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaG inaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGym aiaaicdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRi aaiEdacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaaaaaaa@6FD8@                                              (3.3.3)

Similarly, taking r=3and4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abg2da9iaaiodacaaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlaaisda aaa@3FCE@ in (3.1) and using the respective moments of SBPD, the third and the fourth moment about origin of ZTPSD can be obtained. The variance of ZTPSD (2.3. 1) are thus obtained as

μ 2 = θ 9 +10 θ 8 +58 θ 7 +210 θ 6 +503 θ 5 +760 θ 4 +686 θ 3 +352 θ 2 +96θ+12 θ 2 ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaiMdaaaqcfaOaey4kaSIaaGymaiaaicdacq aH4oqCdaahaaqabKqbGeaacaaI4aaaaKqbakabgUcaRiaaiwdacaaI 4aGaeqiUde3aaWbaaeqajuaibaGaaG4naaaajuaGcqGHRaWkcaaIYa GaaGymaiaaicdacqaH4oqCdaahaaqabKqbGeaacaaI2aaaaKqbakab gUcaRiaaiwdacaaIWaGaaG4maiabeI7aXnaaCaaabeqcfasaaiaaiw daaaqcfaOaey4kaSIaaG4naiaaiAdacaaIWaGaeqiUde3aaWbaaeqa juaibaGaaGinaaaajuaGcqGHRaWkcaaI2aGaaGioaiaaiAdacqaH4o qCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaiodacaaI1aGa aGOmaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG yoaiaaiAdacqaH4oqCcqGHRaWkcaaIXaGaaGOmaaqaaiabeI7aXnaa Caaabeqcfasaaiaaikdaaaqcfa4aaeWaaeaacqaH4oqCdaahaaqabK qbGeaacaaI0aaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabKqb GeaacaaIZaaaaKqbakabgUcaRiaaigdacaaIWaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI3aGaeqiUdeNaey4kaSIa aGOmaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@88FF@

The index of dispersion of ZTPSD (2.3.1) is given by

γ= σ 2 μ = θ 9 +10 θ 8 +58 θ 7 +210 θ 6 +503 θ 5 +760 θ 4 +686 θ 3 +352 θ 2 +96θ+12 θ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 )( θ 5 +5 θ 4 +15 θ 3 +25 θ 2 +20θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTbaacqGH9aqpdaWcaaqaaiabeI7aXnaaCaaabeqcfa saaiaaiMdaaaqcfaOaey4kaSIaaGymaiaaicdacqaH4oqCdaahaaqa bKqbGeaacaaI4aaaaKqbakabgUcaRiaaiwdacaaI4aGaeqiUde3aaW baaeqajuaibaGaaG4naaaajuaGcqGHRaWkcaaIYaGaaGymaiaaicda cqaH4oqCdaahaaqabKqbGeaacaaI2aaaaKqbakabgUcaRiaaiwdaca aIWaGaaG4maiabeI7aXnaaCaaabeqcfasaaiaaiwdaaaqcfaOaey4k aSIaaG4naiaaiAdacaaIWaGaeqiUde3aaWbaaeqajuaibaGaaGinaa aajuaGcqGHRaWkcaaI2aGaaGioaiaaiAdacqaH4oqCdaahaaqabKqb GeaacaaIZaaaaKqbakabgUcaRiaaiodacaaI1aGaaGOmaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGyoaiaaiAdacqaH 4oqCcqGHRaWkcaaIXaGaaGOmaaqaaiabeI7aXnaabmaabaGaeqiUde 3aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWkcaaI0aGaeqiUde3a aWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIXaGaaGimaiabeI 7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG4naiabeI7a XjabgUcaRiaaikdaaiaawIcacaGLPaaadaqadaqaaiabeI7aXnaaCa aabeqcfasaaiaaiwdaaaqcfaOaey4kaSIaaGynaiabeI7aXnaaCaaa beqcfasaaiaaisdaaaqcfaOaey4kaSIaaGymaiaaiwdacqaH4oqCda ahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaikdacaaI1aGaeqiU de3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaaGimai abeI7aXjabgUcaRiaaiAdaaiaawIcacaGLPaaaaaaaaa@A5B3@

It can be easily verified that the ZTPSD is over dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E24@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E26@  and under dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH+aGpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E28@  for θ<(=)> θ = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaiikaiabg2da9iaacMcacqGH+aGpcqaH4oqCdaahaaqa beaacqGHxiIkaaGaeyypa0daaa@4072@ 1.548328 respectively. It is to noted that ZTPLD is over dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E24@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E26@ and under dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH+aGpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E28@  for θ<(=)> θ = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaiikaiabg2da9iaacMcacqGH+aGpcqaH4oqCdaahaaqa beaacqGHxiIkaaGaeyypa0daaa@4072@ 1.258627 respectively.

Estimation of the Parameter

Estimation of parameter of ZTPD

Let x 1 , x 2 ,..., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaaMc8UaaiOlaiaac6cacaGGUaGaaG PaVlaacYcacaWG4bWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@45B8@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  from the ZTPD (2.1.1). The maximum likelihood estimate (MLE) and method of moment estimate (MOME) of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyzam aaCaaabeqcfasaaiabeI7aXbaajuaGdaqadaqaaiqadIhagaqeaiab gkHiTiabeI7aXbGaayjkaiaawMcaaiabgkHiTiqadIhagaqeaiabg2 da9iaaicdaaaa@4305@ , where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean.

Estimation of Parameter of ZTPLD

Maximum Likelihood Estimate (MLE):                                   

The maximum likelihood estimate θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  of ZTPLD is 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaGaamOBaaqaaiabeI7aXbaacqGHsisldaWcaaqaaiaad6ga daqadaqaaiaaikdacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaa aabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI ZaGaeqiUdeNaey4kaSIaaGymaaaacqGHsisldaWcaaqaaiaad6gaca aMc8UabmiEayaaraaabaGaeqiUdeNaey4kaSIaaGymaaaacqGHRaWk daaeWbqaamaalaaabaGaamOzamaaBaaabaGaamiEaaqabaaabaGaam iEaiabgUcaRiabeI7aXjabgUcaRiaaikdaaaaajuaibaGaamiEaiab g2da9iaaigdaaeaacaWGRbaajuaGcqGHris5aiabg2da9iaaicdaaa a@622A@                                     

Where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ 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. [25] showed that the MLE  θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ is consistent and asymptotically normal.

Method of Moment Estimate (MOME): Equating the population mean to the corresponding sample mean, the MOME θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aaceWG4bGbaebacqGHsislcaaIXaaacaGLOaGaayzkaaGaeqiUde3a aWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkdaqadaqaaiaaiodace WG4bGbaebacqGHsislcaaI0aaacaGLOaGaayzkaaGaeqiUde3aaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqadaqaaiqadIhagaqeai abgkHiTiaaiwdaaiaawIcacaGLPaaacqaH4oqCcqGHsislcaaIYaGa eyypa0JaaGimaiaaykW7caaMc8Uaai4oaiqadIhagaqeaiabg6da+i aaigdaaaa@584B@ , where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean. Ghitany et al. [25] showed that the MOME θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ is consistent and asymptotically normal.

Estimation of Parameter of ZTPSD

Maximum Likelihood Estimate (MLE):  The maximum likelihood estimate θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ of ZTPSD is the solution of the following non-linear equation

    x=1 k ( x+2θ+3 ) f x x 2 +( θ+4 )x+( θ 2 +3θ+4 ) n x ¯ θ+1 n( θ 4 10 θ 2 14θ6 ) θ( θ 4 +4 θ 3 +10 θ 2 +7θ+2 ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCae aadaWcaaqaamaabmaabaGaamiEaiabgUcaRiaaikdacqaH4oqCcqGH RaWkcaaIZaaacaGLOaGaayzkaaGaamOzamaaBaaajuaibaGaamiEaa qcfayabaaabaGaamiEamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4k aSYaaeWaaeaacqaH4oqCcqGHRaWkcaaI0aaacaGLOaGaayzkaaGaam iEaiabgUcaRmaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkcaaIZaGaeqiUdeNaey4kaSIaaGinaaGaayjkaiaawM caaaaaaKqbGeaacaWG4bGaeyypa0JaaGymaaqaaiaadUgaaKqbakab ggHiLdGaeyOeI0YaaSaaaeaacaWGUbGaaGPaVlqadIhagaqeaaqaai abeI7aXjabgUcaRiaaigdaaaGaeyOeI0YaaSaaaeaacaWGUbWaaeWa aeaacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbakabgkHiTiaaig dacaaIWaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsisl caaIXaGaaGinaiabeI7aXjabgkHiTiaaiAdaaiaawIcacaGLPaaaae aacqaH4oqCdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqc faOaey4kaSIaaGinaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa Oaey4kaSIaaGymaiaaicdacqaH4oqCdaahaaqabKqbGeaacaaIYaaa aKqbakabgUcaRiaaiEdacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaay zkaaaaaiabg2da9iaaicdaaaa@8D74@           

Where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean. This non-linear equation can be solved by any numerical iteration methods such as Newton-Raphson, Bisection method, Regula-Falsi method etc.

Method of Moment Estimate (MOME): Equating the population mean to the corresponding sample mean, the method of moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  of ZTPSD is the solution of the following non-linear equation

          ( 1 x ¯ ) θ 5 +( 54 x ¯ ) θ 4 +( 1510 x ¯ ) θ 3 +( 257 x ¯ ) θ 2 +( 202 x ¯ )θ+6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIXaGaeyOeI0IabmiEayaaraaacaGLOaGaayzkaaGaeqiUde3a aWbaaeqajuaibaGaaGynaaaajuaGcqGHRaWkdaqadaqaaiaaiwdacq GHsislcaaI0aGabmiEayaaraaacaGLOaGaayzkaaGaeqiUde3aaWba aeqajuaibaGaaGinaaaajuaGcqGHRaWkdaqadaqaaiaaigdacaaI1a GaeyOeI0IaaGymaiaaicdaceWG4bGbaebaaiaawIcacaGLPaaacqaH 4oqCdaahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRmaabmaabaGaaG OmaiaaiwdacqGHsislcaaI3aGabmiEayaaraaacaGLOaGaayzkaaGa eqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqadaqaai aaikdacaaIWaGaeyOeI0IaaGOmaiqadIhagaqeaaGaayjkaiaawMca aiabeI7aXjabgUcaRiaaiAdacqGH9aqpcaaIWaaaaa@67B6@  Where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean?

 Figure 1: Graph of probability mass functions of ZTPD, ZTPLD and ZTPSD for different values of their parameter.

Applications and Goodness of Fit

In this section, an attempt has been made to test the suitability of ZTPD, ZTPLD and ZTPSD 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 [26] have been used as the data of Sri Lanka whereas the data from the survey conducted by Lal [27] and the survey of Kadam Kuan, Patna, conducted in 1975 and referred to by Mishra [28] have been used as the data of India. Further, ZTPD, ZTPLD, and ZTPSD have also been fitted to data on migration. It is obvious from the fitting of ZTPD, ZTPLD and ZTPSD that ZTPSD and ZTPLD are competitive distributions for modeling data from demography (Table A1-A8).

Number of Neonatal Deaths

Observed number of Mothers

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

409

399.7

408.1

408.2

2

88

102.3

89.4

89.2

3

19

17.5 2.2 0.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI3aGaaiOlaiaaiwdaaeaacaaIYaGaaiOlaiaa ikdaaeaacaaIWaGaaiOlaiaaiodaaaGaayzFaaaaaa@3EE9@

19.3
4.1 1.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGymaaaa caGL9baaaaa@3BFB@

19.3
4.1 1.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGOmaaaa caGL9baaaaa@3BFC@

4

5

5

1

Total

522

522.0

522.0

522.0

ML Estimate

θ ^ =0.512047 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaigda caaIYaGaaGimaiaaisdacaaI3aaaaa@3F4B@

θ ^ =4.199697 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaisdacaGGUaGaaGymaiaaiMda caaI5aGaaGOnaiaaiMdacaaI3aaaaa@3F65@

θ ^ =4.655303 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaisdacaGGUaGaaGOnaiaaiwda caaI1aGaaG4maiaaicdacaaIZaaaaa@3F52@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

3.464

0.145

0.113

d.f.

1

2

2

P-value

0.0627

0.9301

0.945

Table A1: 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

ZTPSD

1

71

66.5

72.3

72.1

2

32

35.1

28.4

28.6

3

7

12.3 3.3 0.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaiodaaeaacaaIZaGaaiOlaiaa iodaaeaacaaIWaGaaiOlaiaaiIdaaaGaayzFaaaaaa@3EE9@

10.9
4.1 2.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGymaaqaaiaaikdacaGGUaGaaG4maaaa caGL9baaaaa@3BFE@

10.9
4.1 2.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGymaaqaaiaaikdacaGGUaGaaG4maaaa caGL9baaaaa@3BFE@

4

5

5

3

Total

118

118.0

118.0

118.0

ML Estimate

θ ^ =1.055102 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGimaiaaiwda caaI1aGaaGymaiaaicdacaaIYaaaaa@3F46@

θ ^ =2.049609 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGimaiaaisda caaI5aGaaGOnaiaaicdacaaI5aaaaa@3F56@

θ ^ =2.476545 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGinaiaaiEda caaI2aGaaGynaiaaisdacaaI1aaaaa@3F59@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

0.696

2.274

2.215

d.f.

1

2

2

P-value

0.4041

0.3208

0.3303

Table A2: 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

ZTPSD

1

176

164.3

171.6

171.6

2

44

61.2

51.3

51.3

3

16

15.2 2.8 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI1aGaaiOlaiaaikdaaeaacaaIYaGaaiOlaiaa iIdaaeaacaaIWaGaaiOlaiaaiwdaaaGaayzFaaaaaa@3EEC@

15
4.3 1.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaG4maaqaaiaaigdacaGGUaGaaGioaaaa caGL9baaaaa@3C04@

15.1
4.3 1.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaG4maaqaaiaaigdacaGGUaGaaG4naaaa caGL9baaaaa@3C03@

4

6

5

2

Total

244

244.0

244.0

244.0

ML Estimate

θ ^ =0.744522 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaisda caaI0aGaaGynaiaaikdacaaIYaaaaa@3F50@

θ ^ =2.209411 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOmaiaaicda caaI5aGaaGinaiaaigdacaaIXaaaaa@3F4B@

θ ^ =3.366836 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaiodacaGGUaGaaG4maiaaiAda caaI2aGaaGioaiaaiodacaaI2aaaaa@3F5B@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

7.301

1.882

1.869

d.f.

1

2

2

P-value

0.0069

0.3902

0.3927

Table A3: 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

ZTPSD

1

745

708.9

738.1

738.1

2

212

255.1

214.8

214.9

3

50

61.2

61.3

61.3

4

21

11.0 1.6 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIXaGaaiOlaiaaicdaaeaacaaIXaGaaiOlaiaa iAdaaeaacaaIWaGaaiOlaiaaikdaaaGaayzFaaaaaa@3EE0@

17.2
4.8 1.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaGioaaaa caGL9baaaaa@3C09@

17.2
4.7 1.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaG4naaqaaiaaigdacaGGUaGaaGioaaaa caGL9baaaaa@3C08@

5

7

6

3

Total

1038

1038.0

1038.0

1038.0

ML Estimate

θ ^ =0.719783 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaigda caaI5aGaaG4naiaaiIdacaaIZaaaaa@3F5B@

θ ^ =3.007722 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaiodacaGGUaGaaGimaiaaicda caaI3aGaaG4naiaaikdacaaIYaaaaa@3F4D@

θ ^ =3.466279 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaiodacaGGUaGaaGinaiaaiAda caaI2aGaaGOmaiaaiEdacaaI5aaaaa@3F5D@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

37.046

4.773

4.909

d.f.

2

3

3

P-value

0

0.1892

0.1785

Table A4: 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

ZTPSD

1

683

659.0

674.4

674.7

2

145

177.4

154.1

153.8

3

29

31.8 4.3 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaaIXaGaaiOlaiaaiIdaaeaacaaI0aGaaiOlaiaa iodaaeaacaaIWaGaaiOlaiaaiwdaaaGaayzFaaaaaa@3EED@

34.6
7.7 2.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaG4naaqaaiaaikdacaGGUaGaaGOmaaaa caGL9baaaaa@3C06@

34.7
7.7 2.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaG4naaqaaiaaikdacaGGUaGaaGymaaaa caGL9baaaaa@3C05@

4

11

5

5

Total

873

873.0

873.0

873.0

ML Estimate

θ ^ =0.538402 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaioda caaI4aGaaGinaiaaicdacaaIYaaaaa@3F4E@

θ ^ =4.00231 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaisdacaGGUaGaaGimaiaaicda caaIYaGaaG4maiaaigdaaaa@3E88@

θ ^ =4.462424 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaI0aGaaiOlaiaaisdacaaI2aGaaGOmaiaaisda caaIYaGaaGinaaaa@3F32@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

8.718

5.310

5.463

d.f.

1

2

2

P-value

0.0031

0.0703

0.0651

Table A5: 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

ZTPSD

1

89

76.8

83.4

83.3

2

25

39.9

32.3

32.5

3

11

13.8 3.6 0.7 0.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIZaGaaiOlaiaaiIdaaeaacaaIZaGaaiOlaiaa iAdaaeaacaaIWaGaaiOlaiaaiEdaaeaacaaIWaGaaiOlaiaaikdaaa GaayzFaaaaaa@411A@

12.2
4.5 1.6 1.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGynaaqaaiaaigdacaGGUaGaaGOnaaqa aiaaigdacaGGUaGaaGimaaaacaGL9baaaaa@3E2C@

12.2
4.5 1.6 0.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGynaaqaaiaaigdacaGGUaGaaGOnaaqa aiaaicdacaGGUaGaaGyoaaaacaGL9baaaaa@3E34@

4

6

5

3

6

1

Total

135

135.0

135.0

135.0

ML Estimate

θ ^ =1.038289 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGimaiaaioda caaI4aGaaGOmaiaaiIdacaaI5aaaaa@3F57@

θ ^ =2.089084 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGimaiaaiIda caaI5aGaaGimaiaaiIdacaaI0aaaaa@3F57@

θ ^ =2.525367 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGynaiaaikda caaI1aGaaG4maiaaiAdacaaI3aaaaa@3F56@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

7.90

3.428

3.523

d.f.

1

2

2

P-value

0.0049

0.1801

0.1717

Table A6: 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

ZTPSD

1

567

545.8

561.4

561.5

2

135

162.5

139.7

139.5

3

28

32.3

34.2

34.2

4

11

4.8 0.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaGOnaaaa caGL9baaaaa@3C06@

8.2 2.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGOmaaqaaiaaikdacaGGUaGaaGynaaaa caGL9baaaaa@3C05@

8.2 2.6 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiIdacaGGUaGaaGOmaaqaaiaaikdacaGGUaGaaGOnaaaa caGL9baaaaa@3C06@

5

5

Total

746

746.0

746.0

746.0

ML Estimate

θ ^ =0.595415 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGynaiaaiMda caaI1aGaaGinaiaaigdacaaI1aaaaa@3F55@

θ ^ =3.625737 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaiodacaGGUaGaaGOnaiaaikda caaI1aGaaG4naiaaiodacaaI3aaaaa@3F59@

θ ^ =1.539511 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIXaGaaiOlaiaaiwdacaaIZaGaaGyoaiaaiwda caaIXaGaaGymaaaa@3F31@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

26.855

3.839

3.824

d.f.

2

2

2

P-value

0.0

0.1467

0.1477

Table A7: The number of mothers having at least one neonatal death.

Number of Migrants

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

375

354.0

379.0

378.3

2

143

167.7

137.2

137.8

3

49

53.0

48.4

48.7

4

17

12.5 2.4 0.4 0.1 0.0 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaiwdaaeaacaaIYaGaaiOlaiaa isdaaeaacaaIWaGaaiOlaiaaisdaaeaacaaIWaGaaiOlaiaaigdaae aacaaIWaGaaiOlaiaaicdaaaGaayzFaaaaaa@4336@

16.8
5.7 1.9 0.6 0.4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiwdacaGGUaGaaG4naaqaaiaaigdacaGGUaGaaGyoaaqa aiaaicdacaGGUaGaaGOnaaqaaiaaicdacaGGUaGaaGinaaaacaGL9b aaaaa@4062@

16.8
5.6 1.8 0.6 0.4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiwdacaGGUaGaaGOnaaqaaiaaigdacaGGUaGaaGioaaqa aiaaicdacaGGUaGaaGOnaaqaaiaaicdacaGGUaGaaGinaaaacaGL9b aaaaa@4060@

5

2

6

2

7

1

8

1

Total

590

590.0

590.0

590.0

ML Estimate

θ ^ =0.947486 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGyoaiaaisda caaI3aGaaGinaiaaiIdacaaI2aaaaa@3F5E@

θ ^ =2.284782 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOmaiaaiIda caaI0aGaaG4naiaaiIdacaaIYaaaaa@3F59@

θ ^ =2.722929 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaG4naiaaikda caaIYaGaaGyoaiaaikdacaaI5aaaaa@3F59@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

8.933

1.031

0.912

d.f.

2

3

3

P-value

0.0115

0.7937

0.8225

Table A8: Number of households having at least one migrant according to the number of migrants, reported by Singh & Yadav [29].

Biological Sciences

In this section, an attempt has been made to test the goodness of fit of ZTPD, ZTPLD and ZTPSD on many data- sets relating to biological sciences and it is obvious from the fitting of these distributions that ZTPSD gives much closer fit than ZTPD and ZTPLD in almost all cases. Thus in biological sciences ZTPSD is a better model than ZTPD and ZTPSD to model zero-truncated count data (Table B1-B6).

Number of European Red Mites

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

38

28.7

36.1

35.5

2

17

25.7

20.5

20.8

3

10

15.3

11.2

11.5

4

9

6.9 2.5 0.7 0.2 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGyoaaqaaiaaikdacaGGUaGaaGynaaqa aiaaicdacaGGUaGaaG4naaqaaiaaicdacaGGUaGaaGOmaaqaaiaaic dacaGGUaGaaGymaaaacaGL9baaaaa@4289@

5.9
3.1 1.6 0.8 0.8 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGOnaaqa aiaaicdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaGioaaaacaGL9b aaaaa@405D@

6.1
3.1 1.5 0.8 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiodacaGGUaGaaGymaaqaaiaaigdacaGGUaGaaGynaaqa aiaaicdacaGGUaGaaGioaaqaaiaaicdacaGGUaGaaG4naaaacaGL9b aaaaa@405B@

5

3

6

2

7

1

8

0

Total

80

80.0

80.0

80.0

ML Estimate

θ ^ =1.791615 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaG4naiaaiMda caaIXaGaaGOnaiaaigdacaaI1aaaaa@3F56@

θ ^ =1.185582 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGymaiaaiIda caaI1aGaaGynaiaaiIdacaaIYaaaaa@3F56@

θ ^ =1.539511 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGynaiaaioda caaI5aGaaGynaiaaigdacaaIXaaaaa@3F51@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

9.827

2.467

2.444

d.f.

2

3

3

P-value

0.0073

0.4813

0.4854

Table B1: Number of European red mites on apple leaves, reported by Garman [30].

Number of Yeast Cells Counts per mm Square

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

128

121.3

127.6

127.4

2

37

49.2

40.9

41

3

18

13.3 2.7 0.4 0.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIZaGaaiOlaiaaiodaaeaacaaIYaGaaiOlaiaa iEdaaeaacaaIWaGaaiOlaiaaisdaaeaacaaIWaGaaiOlaiaaigdaaa GaayzFaaaaaa@4111@

12.8 4.0 1.2 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaiIdaaeaacaaI0aGaaiOlaiaa icdaaeaacaaIXaGaaiOlaiaaikdaaeaacaaIWaGaaiOlaiaaiwdaaa GaayzFaaaaaa@4113@

12.9 4.0 1.2 0.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaIYaGaaiOlaiaaiMdaaeaacaaI0aGaaiOlaiaa icdaaeaacaaIXaGaaiOlaiaaikdaaeaacaaIWaGaaiOlaiaaiwdaaa GaayzFaaaaaa@4114@

4

3

5

1

6

0

Total

187

187.0

187.0

187.0

ML Estimate

θ ^ =0.811276 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGioaiaaigda caaIXaGaaGOmaiaaiEdacaaI2aaaaa@3F51@

θ ^ =2.667323 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaiAda caaI3aGaaG4maiaaikdacaaIZaaaaa@3F55@

θ ^ =3.115436 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaiodacaGGUaGaaGymaiaaigda caaI1aGaaGinaiaaiodacaaI2aaaaa@3F4F@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

5.228

1.034

1.013

d.f.

1

1

1

P-value

0.0222

0.3092

0.3141

Table B2: Number of yeast cell counts observed per mm square, reported by Student [31].

Number of Leaf Spot Grade

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

18

14.2

23.0

21.7

2

15

18.7

16.3

16.5

3

10

16.5

11.1

11.6

4

14

10.9

7.3

7.8

5

13

9.7

12.3

12.4

Total

70

70.0

70.0

70.0

ML Estimate

θ ^ =2.639984 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaioda caaI5aGaaGyoaiaaiIdacaaI0aaaaa@3F61@

θ ^ =0.781902 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaiIda caaIXaGaaGyoaiaaicdacaaIYaaaaa@3F53@

θ ^ =1.056454 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGimaiaaiwda caaI2aGaaGinaiaaiwdacaaI0aaaaa@3F51@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

6.311

7.476

5.943

d.f.

3

3

3

P-value

0.0974

0.0582

0.1144

Table B3: The number of leaf spot grade of Ichinose variety of Mulberry, reported by Khurshid [32].

Number of Leaf Spot Grade

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

37

28.5

36.7

35.9

2

16

26.7

21.4

21.8

3

15

16.7

12.0

12.4

4

8

7.8 4.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiEdacaGGUaGaaGioaaqaaiaaisdacaGGUaGaaGOmaaaa caGL9baaaaa@3C09@

6.6

6.8

5

8

7.3

7.1

Total

84

84.0

84.0

84.0

ML Estimate

θ ^ =1.874567 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGioaiaaiEda caaI0aGaaGynaiaaiAdacaaI3aaaaa@3F5E@

θ ^ =1.130211 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGymaiaaioda caaIWaGaaGOmaiaaigdacaaIXaaaaa@3F41@

θ ^ =1.470098 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaigdacaGGUaGaaGinaiaaiEda caaIWaGaaGimaiaaiMdacaaI4aaaaa@3F55@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

8.329

2.477

2.446

d.f.

2

3

3

P-value

0.0155

0.4795

0.4851

Table B4: The number of leaf spot grade of Kokuso-20 variety of Mulberry, reported by Khurshid [32].

Number of Sites with Particles

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

122

115.9

124.8

124.4

2

50

57.4

46.8

47.0

3

18

18.9

17.1

17.2

4

4

4.7 1.1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaG4naaqaaiaaigdacaGGUaGaaGymaaaa caGL9baaaaa@3C01@

6.1 3.2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGymaaqaaiaaiodacaGGUaGaaGOmaaaa caGL9baaaaa@3C00@

6.1 3.3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaiAdacaGGUaGaaGymaaqaaiaaiodacaGGUaGaaG4maaaa caGL9baaaaa@3C01@

5

4

Total

198

198.0

198.0

198.0

ML Estimate

θ ^ =0.990586 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaGyoaiaaiMda caaIWaGaaGynaiaaiIdacaaI2aaaaa@3F5D@

θ ^ =2.18307 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGymaiaaiIda caaIZaGaaGimaiaaiEdaaaa@3E93@

θ ^ =2.614691 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGOnaiaaigda caaI0aGaaGOnaiaaiMdacaaIXaaaaa@3F55@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

2.140

0.510

0.460

d.f.

2

2

2

P-value

0.3430

0.7749

0.7945

Table B5: The number of counts of sites with particles from Immunogold data reported by Mathews & Appleton [33].

Number of Times Hares Caught

Observed Frequency

Expected Frequency

ZTPD

ZTPLD

ZTPSD

1

184

176.6

182.6

182.6

2

55

66.0

55.3

55.3

3

14

16.6 3.1 0.7 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaigdacaaI2aGaaiOlaiaaiAdaaeaacaaIZaGaaiOlaiaa igdaaeaacaaIWaGaaiOlaiaaiEdaaaGaayzFaaaaaa@3EED@

16.4
4.8 1.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaGyoaaaa caGL9baaaaa@3C0A@

16.4
4.8 1.9 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiGaaq aabeqaaiaaisdacaGGUaGaaGioaaqaaiaaigdacaGGUaGaaGyoaaaa caGL9baaaaa@3C0A@

4

4

5

4

Total

261

261.0

261.0

261.0

ML Estimate

θ ^ =0.756171 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaicdacaGGUaGaaG4naiaaiwda caaI2aGaaGymaiaaiEdacaaIXaaaaa@3F53@

θ ^ =2.863957 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpqaaaaaaaaaWdbiaaikdacaGGUaGaaGioaiaaiAda caaIZaGaaGyoaiaaiwdacaaI3aaaaa@3F60@

θ ^ =3.320063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpcaaIZaGaaiOlaiaaiodacaaIYaGaaGimaiaaicda caaI2aGaaG4maaaa@3F29@

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaeqajuaibaGaaGOmaaaaaaa@3947@

2.450

0.610

0.575

d.f.

1

2

2

P-value

0.1175

0.7371

0.7501

Table B6: The number of snowshoe hares counts captured over 7 days, reported by Keith & Meslow [34,35].

Concluding Remarks

In this paper, the nature and behavior of ZTPD, ZTPLD and ZTPSD have been studied by drawing different graphs of their probability functions for the different values of their parameter. A very simple and easy method for finding moments of ZTPSD has been suggested. An attempt has been made to study the goodness of fit of ZTPD, ZTPLD and ZTPSD to count data relating to demography and biological sciences and it has been observed that ZTPSD is a better model than the ZTPD and ZTPLD in almost all data-sets relating to mortality, migration and biological sciences.

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