Research Article
Volume 3 Issue 5  2016
On ZeroTruncation of Poisson, PoissonLindley and PoissonSujatha Distributions and their Applications
Rama Shanker^{1}* and Hagos Fesshaye^{2}
^{1}Department of Statistics, Eritrea Institute of Technology, Eritrea
^{2}Department 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 ZeroTruncation of Poisson, PoissonLindley and PoissonSujatha 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 zerotruncated Poisson distribution (ZTPD), zerotruncated PoissonLindley distribution (ZTPLD) and zerotruncated PoissonSujatha distribution (ZTPSD) have been discussed with graphs of their probability functions for different values of their parameter. Overdispersion, equidispersion and underdispersion 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 zerotruncated data arising from demography, biological sciences and migration has been discussed and observed that in most datasets relating to demography and biological sciences, ZTPSD gives better fit than ZTPD and ZTPLD.
Keywords: Zerotruncated distribution; PoissonLindley distribution; PoissonSujatha distribution; Moments, Estimation of parameter; Goodness of fit
Introduction
Zerotruncated 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 zerocounts.
Let
${P}_{0}\left(x;\theta \right)$
is the original distribution with support non negative positive integers. Then the zerotruncated version of
${P}_{0}\left(x;\theta \right)$
with the support the set of positive integers is given by
$P\left(x;\theta \right)=\frac{{P}_{0}\left(x;\theta \right)}{1{P}_{0}\left(0;\theta \right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=1,2,3,\mathrm{...}$
(1.1)
The PoissonLindley distribution (PLD) having probability mass function (p.m.f.)
${P}_{0}\left(x;\theta \right)=\frac{{\theta}^{2}\left(x+\theta +2\right)}{{\left(\theta +1\right)}^{x+3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,3,\mathrm{...},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0$
(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
$\lambda $
follows Lindley [3] with probability density function (p.d.f.)
$g\left(\lambda ;\theta \right)=\frac{{\theta}^{2}}{\theta +1}\left(1+\lambda \right){e}^{\theta \lambda}\text{\hspace{0.17em}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\lambda >0\text{\hspace{0.17em}}\text{\hspace{0.17em}},\theta >0$
(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 datasets 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 PoissonLindley distribution by compounding Poisson distribution with a two parameter Lindley distribution introduced by Shanker & Mishra [9]. A quasi PoissonLindley 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 PoissonLindley 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 PoissonLindley distribution by compounding Poisson distribution with a new quasi Lindley distribution introduced by Shanker & Amanuel [15].
Recently Shanker [16] has obtained PoissonSujatha distribution (PSD) having p.m.f.
${P}_{0}\left(x;\theta \right)=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\frac{{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)}{{\left(\theta +1\right)}^{x+3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=0,1,2,3,\mathrm{...},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0$
(1.4)
to model count data in different fields of knowledge. The PSD arises from the Poisson distribution when its parameter
$\lambda $
follows Shanker R [17] with probability density function (p.d.f.)
$g\left(\lambda ;\theta \right)=\frac{{\theta}^{3}}{{\theta}^{2}+\theta +2}\left(1+\lambda +{\lambda}^{2}\right){e}^{\theta \text{\hspace{0.17em}}\lambda}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\text{\hspace{0.17em}}\lambda >0\text{\hspace{0.17em}}\text{\hspace{0.17em}},\theta >0$
(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 sizebiased and zerotruncated version of PSD and discussed their properties, estimation of parameter and applications in different fields of knowledge.
In this paper, the nature of zerotruncated Poisson distribution (ZTPD), zerotruncated PoissonLindley distribution (ZTPLD) and zerotruncated PoissonSujatha 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.
ZeroTruncated Poisson, PoissonLindley and PoissonSujatha Distributions
Zerotruncated poisson distribution (ZTPD)
Using (1.1) and the p.m.f. of Poisson distribution, the p.m.f. of zerotruncated Poisson distribution (ZTPD) given by
${P}_{1}\left(x;\theta \right)=\frac{{\theta}^{x}}{\left({e}^{\theta}1\right)\text{\hspace{0.17em}}x!}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=1,2,3,\mathrm{....},\text{\hspace{0.17em}}\theta >0$
(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.
Zerotruncated poissonlindley distribution (ZTPLD)
Using (1.1) and (1.2), the p.m.f. of zerotruncated Poisson Lindley distribution (ZTPLD) given by
${P}_{2}\left(x;\theta \right)=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\frac{x+\theta +2}{{\left(\theta +1\right)}^{x}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=1,2,3,\mathrm{....},\text{\hspace{0.17em}}\theta >0$
(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 zerocounts and showed that in demography and biological sciences ZTPLD gives better fit than ZTPD while in social sciences ZTPD gives better fit than ZTPLD.
ZeroTruncated PoissonSujatha Distribution (ZTPSD)
Using (1.1) and (1.4), the p.m.f. of zerotruncated PoissonSujatha distribution (ZTPSD) can be obtained as
${P}_{3}\left(x;\theta \right)=\frac{{\theta}^{3}}{{\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2}\frac{{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)}{{\left(\theta +1\right)}^{x}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=1,2,3,\mathrm{...},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0$
(2.3.1)
The ZTPSD can also arise from the sizebiased Poisson distribution (SBPD) with p.m.f.
$g\left(x\lambda \right)=\frac{{e}^{\lambda}{\lambda}^{x1}}{\left(x1\right)!}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=1,2,3,\mathrm{...}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\lambda >0$
(2.3.2)
When its parameter
$\lambda $
follows a distribution having p.d.f.
$h\left(\lambda ;\theta \right)=\frac{{\theta}^{3}}{{\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2}\left[{\left(\theta +1\right)}^{2}{\lambda}^{2}+\left(\theta +1\right)\left(\theta +3\right)\lambda +\left({\theta}^{2}+3\theta +4\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}$
$,\lambda >0,\text{\hspace{0.17em}}\theta >0$
(2.3.3)
The p.m.f. of ZTPSD is thus can be obtained as
$P\left(x;\theta \right)={\displaystyle \underset{0}{\overset{\infty}{\int}}g\left(x\lambda \right)}\cdot h\left(\lambda ;\theta \right)d\lambda $
$={\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{{e}^{\lambda}{\lambda}^{x1}}{\left(x1\right)!}}\cdot \frac{{\theta}^{3}}{{\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2}\left[{\left(\theta +1\right)}^{2}{\lambda}^{2}+\left(\theta +1\right)\left(\theta +3\right)\lambda +\left({\theta}^{2}+3\theta +4\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(2.3.4)
$=\frac{{\theta}^{3}}{\left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)\left(x1\right)!}{\displaystyle \underset{0}{\overset{\infty}{\int}}{e}^{\left(\theta +1\right)\lambda}}\cdot \left[\begin{array}{l}{\left(\theta +1\right)}^{2}{\lambda}^{x+21}+\left(\theta +1\right)\left(\theta +3\right){\lambda}^{x+11}\\ +\left({\theta}^{2}+3\theta +4\right){\lambda}^{x1}\end{array}\right]d\lambda $
$=\frac{{\theta}^{3}}{\left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)}\left[\frac{\left(x+1\right)x}{{\left(\theta +1\right)}^{x}}+\frac{\left(\theta +3\right)x}{{\left(\theta +1\right)}^{x}}+\frac{{\theta}^{2}+3\theta +4}{{\left(\theta +1\right)}^{x}}\right]$
$=\frac{{\theta}^{3}}{\left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)}\frac{{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)}{{\left(\theta +1\right)}^{x}}$
$x=1,2,3,\mathrm{...},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0$
which is the p.m.f. of ZTPSD with parameter
$\theta $
.
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
$r$
^{th} moment about origin of ZTPSD (2.3.1) can be obtained as
${\mu}_{r}{}^{\prime}=E\left[E\left({X}^{r}\lambda \right)\right]$
$=\frac{{\theta}^{3}}{{\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left[{\displaystyle \sum _{x=1}^{\infty}{x}^{r}}\frac{{e}^{\lambda}{\lambda}^{x1}}{\left(x1\right)!}\right]}\cdot \left[\begin{array}{l}{\left(\theta +1\right)}^{2}{\lambda}^{2}+\left(\theta +1\right)\left(\theta +3\right)\lambda \\ +\left({\theta}^{2}+3\theta +4\right)\end{array}\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(3.1)
Clearly the expression under the bracket in (3.1) is the
$r$
^{th} moment about origin of the SBPD. Taking
$r=1$
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
${\mu}_{1}{}^{\prime}=\frac{{\theta}^{3}}{{\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left(\lambda +1\right)}\left[\begin{array}{l}{\left(\theta +1\right)}^{2}{\lambda}^{2}+\left(\theta +1\right)\left(\theta +3\right)\lambda \\ +\left({\theta}^{2}+3\theta +4\right)\end{array}\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
$=\frac{{\theta}^{5}+5{\theta}^{4}+15{\theta}^{3}+25{\theta}^{2}+20\theta +6}{\theta \left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)}$
(3.2)
Again taking
$r=2$
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
${\mu}_{2}{}^{\prime}=\frac{{\theta}^{3}}{{\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left({\lambda}^{2}+3\lambda +1\right)}\left[\begin{array}{l}{\left(\theta +1\right)}^{2}{\lambda}^{2}+\left(\theta +1\right)\left(\theta +3\right)\lambda \\ +\left({\theta}^{2}+3\theta +4\right)\end{array}\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
$=\frac{{\theta}^{6}+7{\theta}^{5}+27{\theta}^{4}+73{\theta}^{3}+112{\theta}^{2}+84\theta +24}{{\theta}^{2}\left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)}$
(3.3.3)
Similarly, taking
$r=3\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}4$
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
${\mu}_{2}=\frac{{\theta}^{9}+10{\theta}^{8}+58{\theta}^{7}+210{\theta}^{6}+503{\theta}^{5}+760{\theta}^{4}+686{\theta}^{3}+352{\theta}^{2}+96\theta +12}{{\theta}^{2}{\left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)}^{2}}$
The index of dispersion of ZTPSD (2.3.1) is given by
$\gamma =\frac{{\sigma}^{2}}{\mu}=\frac{{\theta}^{9}+10{\theta}^{8}+58{\theta}^{7}+210{\theta}^{6}+503{\theta}^{5}+760{\theta}^{4}+686{\theta}^{3}+352{\theta}^{2}+96\theta +12}{\theta \left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)\left({\theta}^{5}+5{\theta}^{4}+15{\theta}^{3}+25{\theta}^{2}+20\theta +6\right)}$
It can be easily verified that the ZTPSD is over dispersed
$\left(\mu <{\sigma}^{2}\right)$
, equidispersed
$\left(\mu ={\sigma}^{2}\right)$
and under dispersed
$\left(\mu >{\sigma}^{2}\right)$
for
$\theta <(=)>{\theta}^{\ast}=$
1.548328 respectively. It is to noted that ZTPLD is over dispersed
$\left(\mu <{\sigma}^{2}\right)$
, equidispersed
$\left(\mu ={\sigma}^{2}\right)$
and under dispersed
$\left(\mu >{\sigma}^{2}\right)$
for
$\theta <(=)>{\theta}^{\ast}=$
1.258627 respectively.
Estimation of the Parameter
Estimation of parameter of ZTPD
Let
${x}_{1},{x}_{2},\text{\hspace{0.17em}}\mathrm{...}\text{\hspace{0.17em}},{x}_{n}$
be a random sample of size
$n$
from the ZTPD (2.1.1). The maximum likelihood estimate (MLE) and method of moment estimate (MOME) of
$\theta $
of ZTPD (2.1.1) is given by the solution of the following non linear equation
${e}^{\theta}\left(\overline{x}\theta \right)\overline{x}=0$
, where
$\overline{x}$
is the sample mean.
Estimation of Parameter of ZTPLD
Maximum Likelihood Estimate (MLE):
The maximum likelihood estimate
$\widehat{\theta}$
of
$\theta $
of ZTPLD is the solution of the following nonlinear equation
$\frac{2n}{\theta}\frac{n\left(2\theta +3\right)}{{\theta}^{2}+3\theta +1}\frac{n\text{\hspace{0.17em}}\overline{x}}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{{f}_{x}}{x+\theta +2}}=0$
Where
$\overline{x}$
is the sample mean. This nonlinear equation can be solved by any numerical iteration methods such as NewtonRaphson method, Bisection method, RegulaFalsi method etc. Ghitany et al. [25] showed that the MLE
$\widehat{\theta}$
of
$\theta $
is consistent and asymptotically normal.
Method of Moment Estimate (MOME): Equating the population mean to the corresponding sample mean, the MOME
$\tilde{\theta}$
of
$\theta $
of ZTPLD (2.2.1) is the solution of the following cubic equation
$\left(\overline{x}1\right){\theta}^{3}+\left(3\overline{x}4\right){\theta}^{2}+\left(\overline{x}5\right)\theta 2=0\text{\hspace{0.17em}}\text{\hspace{0.17em}};\overline{x}>1$
, where
$\overline{x}$
is the sample mean. Ghitany et al. [25] showed that the MOME
$\tilde{\theta}$
of
$\theta $
is consistent and asymptotically normal.
Estimation of Parameter of ZTPSD
Maximum Likelihood Estimate (MLE): The maximum likelihood estimate
$\widehat{\theta}$
of
$\theta $
of ZTPSD is the solution of the following nonlinear equation
$\sum _{x=1}^{k}\frac{\left(x+2\theta +3\right){f}_{x}}{{x}^{2}+\left(\theta +4\right)x+\left({\theta}^{2}+3\theta +4\right)}}\frac{n\text{\hspace{0.17em}}\overline{x}}{\theta +1}\frac{n\left({\theta}^{4}10{\theta}^{2}14\theta 6\right)}{\theta \left({\theta}^{4}+4{\theta}^{3}+10{\theta}^{2}+7\theta +2\right)}=0$
Where
$\overline{x}$
is the sample mean. This nonlinear equation can be solved by any numerical iteration methods such as NewtonRaphson, Bisection method, RegulaFalsi method etc.
Method of Moment Estimate (MOME): Equating the population mean to the corresponding sample mean, the method of moment estimate (MOME)
$\tilde{\theta}$
of
$\theta $
of ZTPSD is the solution of the following nonlinear equation
$\left(1\overline{x}\right){\theta}^{5}+\left(54\overline{x}\right){\theta}^{4}+\left(1510\overline{x}\right){\theta}^{3}+\left(257\overline{x}\right){\theta}^{2}+\left(202\overline{x}\right)\theta +6=0$
Where
$\overline{x}$
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 datasets considered here are the data of Sri Lanka and India. The datasets 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 A1A8).
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 
$\begin{array}{l}17.5\\ 2.2\\ 0.3\end{array}\}$

19.3
$\begin{array}{l}4.1\\ 1.1\end{array}\}$

19.3
$\begin{array}{l}4.1\\ 1.2\end{array}\}$

4 
5 
5 
1 
Total 
522 
522.0 
522.0 
522.0 
ML Estimate 

$\widehat{\theta}=0.512047$

$\widehat{\theta}=4.199697$

$\widehat{\theta}=4.655303$

${\chi}^{2}$


3.464 
0.145 
0.113 
d.f. 

1 
2 
2 
Pvalue 

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 
$\begin{array}{l}12.3\\ 3.3\\ 0.8\end{array}\}$

10.9
$\begin{array}{l}4.1\\ 2.3\end{array}\}$

10.9
$\begin{array}{l}4.1\\ 2.3\end{array}\}$

4 
5 
5 
3 
Total 
118 
118.0 
118.0 
118.0 
ML Estimate 

$\widehat{\theta}=1.055102$

$\widehat{\theta}=2.049609$

$\widehat{\theta}=2.476545$

${\chi}^{2}$


0.696 
2.274 
2.215 
d.f. 

1 
2 
2 
Pvalue 

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 
$\begin{array}{l}15.2\\ 2.8\\ 0.5\end{array}\}$

15
$\begin{array}{l}4.3\\ 1.8\end{array}\}$

15.1
$\begin{array}{l}4.3\\ 1.7\end{array}\}$

4 
6 
5 
2 
Total 
244 
244.0 
244.0 
244.0 
ML Estimate 

$\widehat{\theta}=0.744522$

$\widehat{\theta}=2.209411$

$\widehat{\theta}=3.366836$

${\chi}^{2}$


7.301 
1.882 
1.869 
d.f. 

1 
2 
2 
Pvalue 

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 
$\begin{array}{l}11.0\\ 1.6\\ 0.2\end{array}\}$

17.2
$\begin{array}{l}4.8\\ 1.8\end{array}\}$

17.2
$\begin{array}{l}4.7\\ 1.8\end{array}\}$

5 
7 
6 
3 
Total 
1038 
1038.0 
1038.0 
1038.0 
ML Estimate 

$\widehat{\theta}=0.719783$

$\widehat{\theta}=3.007722$

$\widehat{\theta}=3.466279$

${\chi}^{2}$


37.046 
4.773 
4.909 
d.f. 

2 
3 
3 
Pvalue 

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 
$\begin{array}{l}31.8\\ 4.3\\ 0.5\end{array}\}$

34.6
$\begin{array}{l}7.7\\ 2.2\end{array}\}$

34.7
$\begin{array}{l}7.7\\ 2.1\end{array}\}$

4 
11 
5 
5 
Total 
873 
873.0 
873.0 
873.0 
ML Estimate 

$\widehat{\theta}=0.538402$

$\widehat{\theta}=4.00231$

$\widehat{\theta}=4.462424$

${\chi}^{2}$


8.718 
5.310 
5.463 
d.f. 

1 
2 
2 
Pvalue 

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 
$\begin{array}{l}13.8\\ 3.6\\ 0.7\\ 0.2\end{array}\}$

12.2
$\begin{array}{l}4.5\\ 1.6\\ 1.0\end{array}\}$

12.2
$\begin{array}{l}4.5\\ 1.6\\ 0.9\end{array}\}$

4 
6 
5 
3 
6 
1 
Total 
135 
135.0 
135.0 
135.0 
ML Estimate 

$\widehat{\theta}=1.038289$

$\widehat{\theta}=2.089084$

$\widehat{\theta}=2.525367$

${\chi}^{2}$


7.90 
3.428 
3.523 
d.f. 

1 
2 
2 
Pvalue 

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 
$\begin{array}{l}4.8\\ 0.6\end{array}\}$

$\begin{array}{l}8.2\\ 2.5\end{array}\}$

$\begin{array}{l}8.2\\ 2.6\end{array}\}$

5 
5 
Total 
746 
746.0 
746.0 
746.0 
ML Estimate 

$\widehat{\theta}=0.595415$

$\widehat{\theta}=3.625737$

$\widehat{\theta}=1.539511$

${\chi}^{2}$


26.855 
3.839 
3.824 
d.f. 

2 
2 
2 
Pvalue 

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 
$\begin{array}{l}12.5\\ 2.4\\ 0.4\\ 0.1\\ 0.0\end{array}\}$

16.8
$\begin{array}{l}5.7\\ 1.9\\ 0.6\\ 0.4\end{array}\}$

16.8
$\begin{array}{l}5.6\\ 1.8\\ 0.6\\ 0.4\end{array}\}$

5 
2 
6 
2 
7 
1 
8 
1 
Total 
590 
590.0 
590.0 
590.0 
ML Estimate 

$\widehat{\theta}=0.947486$

$\widehat{\theta}=2.284782$

$\widehat{\theta}=2.722929$

${\chi}^{2}$


8.933 
1.031 
0.912 
d.f. 

2 
3 
3 
Pvalue 

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 zerotruncated count data (Table B1B6).
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 
$\begin{array}{l}6.9\\ 2.5\\ 0.7\\ 0.2\\ 0.1\end{array}\}$

5.9
$\begin{array}{l}3.1\\ 1.6\\ 0.8\\ 0.8\end{array}\}$

6.1
$\begin{array}{l}3.1\\ 1.5\\ 0.8\\ 0.7\end{array}\}$

5 
3 
6 
2 
7 
1 
8 
0 
Total 
80 
80.0 
80.0 
80.0 
ML Estimate 

$\widehat{\theta}=1.791615$

$\widehat{\theta}=1.185582$

$\widehat{\theta}=1.539511$

${\chi}^{2}$


9.827 
2.467 
2.444 
d.f. 

2 
3 
3 
Pvalue 

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 
$\begin{array}{l}13.3\\ 2.7\\ 0.4\\ 0.1\end{array}\}$

$\begin{array}{l}12.8\\ 4.0\\ 1.2\\ 0.5\end{array}\}$

$\begin{array}{l}12.9\\ 4.0\\ 1.2\\ 0.5\end{array}\}$

4 
3 
5 
1 
6 
0 
Total 
187 
187.0 
187.0 
187.0 
ML Estimate 

$\widehat{\theta}=0.811276$

$\widehat{\theta}=2.667323$

$\widehat{\theta}=3.115436$

${\chi}^{2}$


5.228 
1.034 
1.013 
d.f. 

1 
1 
1 
Pvalue 

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 

$\widehat{\theta}=2.639984$

$\widehat{\theta}=0.781902$

$\widehat{\theta}=1.056454$

${\chi}^{2}$


6.311 
7.476 
5.943 
d.f. 

3 
3 
3 
Pvalue 

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 
$\begin{array}{l}7.8\\ 4.2\end{array}\}$

6.6 
6.8 
5 
8 
7.3 
7.1 
Total 
84 
84.0 
84.0 
84.0 
ML Estimate 

$\widehat{\theta}=1.874567$

$\widehat{\theta}=1.130211$

$\widehat{\theta}=1.470098$

$${\chi}^{2}$$


8.329 
2.477 
2.446 
d.f. 

2 
3 
3 
Pvalue 

0.0155 
0.4795 
0.4851 
Table B4: The number of leaf spot grade of Kokuso20 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 
$\begin{array}{l}4.7\\ 1.1\end{array}\}$

$\begin{array}{l}6.1\\ 3.2\end{array}\}$

$\begin{array}{l}6.1\\ 3.3\end{array}\}$

5 
4 
Total 
198 
198.0 
198.0 
198.0 
ML Estimate 

$\widehat{\theta}=0.990586$

$\widehat{\theta}=2.18307$

$\widehat{\theta}=2.614691$

${\chi}^{2}$


2.140 
0.510 
0.460 
d.f. 

2 
2 
2 
Pvalue 

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 
$\begin{array}{l}16.6\\ 3.1\\ 0.7\end{array}\}$

16.4
$\begin{array}{l}4.8\\ 1.9\end{array}\}$

16.4
$\begin{array}{l}4.8\\ 1.9\end{array}\}$

4 
4 
5 
4 
Total 
261 
261.0 
261.0 
261.0 
ML Estimate 

$\widehat{\theta}=0.756171$

$\widehat{\theta}=2.863957$

$\widehat{\theta}=3.320063$

${\chi}^{2}$


2.450 
0.610 
0.575 
d.f. 

1 
2 
2 
Pvalue 

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 datasets relating to mortality, migration and biological sciences.
References
 Sankaran M (1970) The discrete PoissonLindley distribution. Biometrics 26: 145149.
 Shanker R, Hagos F (2015) On PoissonLindley distribution and Its Applications to Biological Sciences. Biometrics and Biostatistics International Journal 2(4): 15.
 Lindley DV (1958) Fiducial distributions and Bayes’ Theorem. Journal of the Royal Statistical Society 20(1): 102107.
 Ghitany ME, Atieh B, Nadarajah S (2008a) Lindley distribution and Its Applications. Mathematics Computation and Simulation 78(4): 493506.
 Shanker R, Hagos F, Sujatha S (2015) On modeling of Lifetimes data using exponential and Lindley distributions. Biometrics and Biostatistics International Journal 2(5): 19.
 Ghitany ME, Al Mutairi DK (2009) Estimation Methods for the discrete PoissonLindley distribution. Journal of Statistical Computation and Simulation 79(1): 19.
 Mishra A (2014) A twoparameter PoissonLindley distribution. International Journal of Statistics and Systems 9(1): 7985.
 Shanker R, Mishra A (2013 a) A twoparameter Lindley distribution. Statistics in Transition new Series 14(1): 4556.
 Shanker R, Mishra A (2015) A quasi PoissonLindley distribution. Accepted in Journal of Indian Statistical Association.
 Shanker R, Mishra A (2013 b) A quasi Lindley distribution. African journal of Mathematics and Computer Science Research 6(4): 6471.
 Shanker R, Sharma S, Shanker R (2012) A Discrete twoParameter Poisson Lindley Distribution. Journal of Ethiopian Statistical Association 21: 1522.
 Shanker R, Sharma S, Shanker R (2013) A twoparameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics 4(2): 363368.
 Shanker R, Tekie AL (2013) A new quasi PoissonLindley distribution. International Journal of Statistics and Systems 9(1): 8794.
 Shanker R, Amanuel AG (2013) A new quasi Lindley distribution. International Journal of Statistics and Systems. 6(4): 143156.
 Shanker R (2016 b) The discrete PoissonSujatha distribution. International Journal of Probability and Statistics 5(1): 19.
 Shanker R (2016 a) Sujatha distribution and Its Applications. Accepted for publication in “Statistics in Transitionnew Series”.
 Shanker R, Hagos F (2016 a) SizeBiased PoissonSujatha distribution with Applications, Communicated.
 Shanker R, Hagos F (2016 b) Zerotruncated PoissonSujatha distribution with Applications. Communicated.
 Plackett RL (1953) The truncated Poissondistribution. Biometrics 9(4): 485488.
 David FN, Johnson NL (1952) The truncated Poisson. Biometrics 8: 275285.
 Cohen AC (1960 a) An extension of a truncated Poisson distribution. Biometrics 16(3): 446450.
 Cohen AC (1960 b) Estimation in a truncated Poisson distribution when zeros and some ones are missing. Journal of American Statistical Association 55: 342348.
 Tate RF, Goen RL (1958) MVUE for the truncated Poisson distribution. Annals of Mathematical Statistics 29: 755765.
 Ghitany ME, Al Mutairi DK, Nadarajah S (2008b) Zerotruncated PoissonLindley distribution and its Applications. Mathematics and Computers in Simulation 79(3): 279287.
 Shanker R, Hagos F, Sujatha S, Abrehe Y (2015) On Zerotruncation of Poisson and PoissonLindley distributins and Their Applications. Biometrics and Biostatistics International Journal 2(6): 114
 Meegama SA (1980) Socioeconomic determinants of infant and child mortality in Sri Lanka, an analysis of postwar experience. International Statistical Institute 55.
 Lal DN (1955) Patna in 1955: A Demographic Sample Survey, Demographic Research Center, Department of Statistics, Patna University, Patna, India.
 Mishra A (1979) Generalizations of some discrete distributions. Patna University, Patna, India.
 Singh SN, Yadav RC (1971) Trends in rural outmigration at household level. Rural Demography 8: 5361.
 Garman P (1923) The European red mites in Connecticut apple orchards, Connecticut. Agri Exper Station Bull 252: 103125.
 Student (1907) On the error of counting with a haemacytometer. Biometrika 5(3): 351360.
 Khursid AM (2008) On sizebiased Poisson distribution and Its use in zerotruncated case. Journal of the Korean Society for Industrial and Applied Mathematics 12(3): 153160.
 Mathews JNS, Appleton DR (1993) An application of the truncated Poisson distribution to Immunogold assay. Biometrics 49(2): 617621.
 Keith LB, Meslow EC (1968) Trap response by snowshoe hares. The Journal of Wildlife Management 32(4): 795801.