ISSN: 2378315X BBIJ
Biometrics & Biostatistics International Journal
Research Article
Volume 2 Issue 6  2014
On ZeroTruncation of Poisson and PoissonLindley Distributions and Their Applications
Rama Shanker^{1}*, Hagos Fesshaye^{}2, Sujatha Selvaraj^{3} and Abrehe Yemane^{4}
^{1}Department of Statistics, Eritrea Institute of Technology, Eritrea
^{2}Department of Economics, College of Business and Economics, Eritrea
^{3}Department of Banking and Finance, Jimma University, Ethiopia
^{4}Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: June 23, 2015  Published: July 22, 2015
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H, Selvaraj S, Yemane A (2015) On ZeroTruncation of Poisson and PoissonLindley Distributions and Their Applications. Biom Biostat Int J 2(6): 00045. DOI: 10.15406/bbij.2015.02.00045
Abstract
In this paper, a general expression for the $r$
th factorial moment of zerotruncated PoissonLindley distribution (ZTPLD) has been obtained and hence the first four moments about origin has been given. A very simple and alternative method for finding moments of ZTPLD has also been suggested. The expression for the moment generating function of ZTPLD has been obtained. Both ZTPD (Zerotruncated Poisson distribution) and ZTPLD have been fitted using maximum likelihood estimate to a number of data sets from demography, biological sciences and social sciences and it has been observed that in most cases ZTPLD gives much closer fits than ZTPD while in some cases ZTPD gives much closer fits than ZTPLD.
Keywords: PoissonLindley distribution; Zerotruncated distribution; Moments; Estimation of parameter; Goodness of fit
Abbreviations
ZTPLD: ZeroTruncated PoissonLindley Distribution; ZTPD: ZeroTruncated Poisson Distribution; PLD: PoissonLindley Distribution; PMF: Probability Mass Function; PDF: Probability Density Function; SBPD: SizeBiased Poisson Distribution; MLE: Maximum Likelihood Estimate; MOME: Method of Moment Estimate; MVUE: Minimum Variance Unbiased Estimation; SBPD: SizeBiased Poisson Distribution
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) with parameter$$\theta $$
and 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 et al. [2] have done an extensive study on its applications to Biological Sciences and found that PLD provides a better fit than Poisson distribution to almost all biological science data. The PSD arises from the Poisson distribution when its parameter $\lambda $
follows Lindley distribution [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. Recently, Shanker et al. [2] showed that (1.3) is not always a better model than the exponential distribution for modeling lifetimes data. In fact, Shanker et al. [2] has done a very extensive and comparative study on modeling of lifetimes data using exponential and Lindley distributions and discussed various lifetimes examples to show the superiority of Lindley over exponential and that of exponential over Lindley distribution. The PLD has been extensively studied by Sankaran [1] and Ghitany & Mutairi [5] and they have discussed its various properties. The Lindley distribution and the PLD has been generalized by many researchers. Shanker et al. [6] obtained a two parameter PoissonLindley distribution by compounding Poisson distribution with a two parameter Lindley distribution introduced by Shanker et al. [7]. A quasi PoissonLindley distribution has been introduced by Shanker et al. [8] by compounding Poisson distribution with a quasi Lindley distribution introduced by Shanker et al. [9]. Shanker et al. [10] obtained a discrete two parameter PoissonLindley distribution by mixing Poisson distribution with a two parameter Lindley distribution for modeling waiting and survival times data introduced by Shanker et al. [11]. Further, Shanker et al. [12] obtained a new quasi PoissonLindley distribution by compounding Poisson distribution with a new quasi Lindley distribution introduced by Shanker et al. [13,14].
In this paper, the nature of zerotruncated Poisson distribution (ZTPD) and zerotruncated PoissonLindley distribution (ZTPLD) has been compared and studied using graphs for different values of their parameter. A general expression for the $r$th factorial moment of ZTPLD has been obtained and the first four moment about origin has been given. A very simple and easy method for finding moments of ZTPLD has been suggested. Both ZTPD and ZTPLD have been fitted to a number of data sets from different fields to study their goodness of fits and superiority of one over the other.
ZeroTruncated Poisson and PoissonLindley Distribution
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 [15] and David et al. [16] to model count data that structurally excludes zero counts. An extension of a truncated Poisson distribution and estimation in a truncated Poisson distribution when zeros and some ones are missing has been discussed by Cohen [17,18]. Tate et al. [19] has discussed minimum variance unbiased estimation (MVUE) for the truncated Poisson distribution.
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. [20] to model count data for the missing zeros. It has been shown by Ghitany et al. [20] that ZTPLD can also arise from the sizebiased Poisson distribution (SBPD) with p.m.f.
$f\left(x\lambda \right)=\frac{{e}^{\lambda}{\lambda}^{x1}}{\left(x1\right)!}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=1,2,3,\mathrm{...},\lambda >0$
(2.2.2)
when its parameter $\lambda $
follows a distribution having p.d.f.
$h\left(\lambda ;\theta \right)=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \lambda}\text{\hspace{0.17em}}\text{\hspace{0.17em}};\lambda >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta >0$
(2.2.3)
Thus the p.m.f. of ZTPLD is obtained as
$P\left(X=x\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}f\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}^{2}}{{\theta}^{2}+3\theta +1}\left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \lambda}d\lambda $
(2.2.4)
$=\frac{{\theta}^{2}}{\left({\theta}^{2}+3\theta +1\right)\left(x1\right)!}{\displaystyle \underset{0}{\overset{\infty}{\int}}{e}^{\left(\theta +1\right)\lambda}}\cdot \left[\left(\theta +1\right){\lambda}^{x}+\left(\theta +2\right){\lambda}^{x1}\right]d\lambda $
$=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\left[\frac{x}{{\left(\theta +1\right)}^{x}}+\frac{\left(\theta +2\right)}{{\left(\theta +1\right)}^{x}}\right]$
$=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\frac{x+\theta +2}{{\left(\theta +1\right)}^{x}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}};x=1,2,3,\mathrm{...},\theta >0$
which is the p.m.f. of ZTPLD with parameter$\theta $
.
To study the nature and behaviors of ZTPD and ZTPLD for different values of its parameter, a number of graphs of their probability densities have been drawn and presented in Figure 1.
Moments and related measures
Moments of ZTPD
The $r$
th factorial moment of the ZTPD (2.1.1) can be obtained as
${\mu}_{\left(r\right)}{}^{\prime}=E\left[{X}^{\left(r\right)}\right]=\frac{1}{{e}^{\theta}1}{\displaystyle \sum _{x=1}^{\infty}{x}^{\left(r\right)}}\frac{{\theta}^{x}}{x!}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$
, where ${X}^{\left(r\right)}=X(X1)(X2)\mathrm{...}(Xr+1)$
$=\frac{{\theta}^{r}}{{e}^{\theta}1}{\displaystyle \sum _{x=r}^{\infty}\frac{{\theta}^{xr}}{\left(xr\right)!}=\frac{{\theta}^{r}\text{\hspace{0.17em}}{e}^{\theta}}{{e}^{\theta}1}}$
(3.1.1)
Substituting $r=1,2,3,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{4}$
in (3.1.1), the first four factorial moments can be obtained and, therefore, using the relationship between factorial moments and moments about origin, the first four moments about origin of ZTPD (2.1.1) are obtained as
${\mu}_{1}{}^{\prime}=\frac{\theta \text{\hspace{0.17em}}{e}^{\theta}}{{e}^{\theta}1}$
${\mu}_{2}{}^{\prime}=\frac{\theta \text{\hspace{0.17em}}{e}^{\theta}}{{e}^{\theta}1}\left(\theta +1\right)$
${\mu}_{3}{}^{\prime}=\frac{\theta \text{\hspace{0.17em}}{e}^{\theta}}{{e}^{\theta}1}\left({\theta}^{2}+3\theta +1\right)$
${\mu}_{3}{}^{\prime}=\frac{\theta \text{\hspace{0.17em}}{e}^{\theta}}{{e}^{\theta}1}\left({\theta}^{3}+6{\theta}^{2}+7\theta +1\right)$
Generating Function: The probability generating function of the ZTPD (2.1.1) is obtained as
${P}_{X}\left(t\right)=E\left({t}^{X}\right)=\frac{1}{{e}^{\theta}1}{\displaystyle \sum _{x=1}^{\infty}\frac{{\left(\theta t\right)}^{x}}{x!}}=\frac{1}{{e}^{\theta}1}\left[{\displaystyle \sum _{x=0}^{\infty}\frac{{\left(\theta t\right)}^{x}}{x!}1}\right]=\frac{{e}^{\theta t}1}{{e}^{\theta}1}$
The moment generating function of the ZTPD (2.1.1) is thus given by
${M}_{X}\left(t\right)=E\left({e}^{tX}\right)=\frac{{e}^{\theta \text{\hspace{0.17em}}{e}^{t}}1}{{e}^{\theta}1}$
Moments of ZTPLD
The$r$
th factorial moment of the ZTPLD (2.2.1) can be obtained as
${\mu}_{\left(r\right)}{}^{\prime}=E\left[{X}^{\left(r\right)}\lambda \right]$
, where ${X}^{\left(r\right)}=X(X1)(X2)\mathrm{...}(Xr+1)$
Using (2.2.4), we get
${\mu}_{\left(r\right)}{}^{\prime}={\displaystyle \underset{0}{\overset{\infty}{\int}}\left[{\displaystyle \sum _{x=1}^{\infty}{x}^{\left(r\right)}\frac{{e}^{\lambda}{\lambda}^{x1}}{\left(x1\right)!}}\right]}\cdot \frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
$={\displaystyle \underset{0}{\overset{\infty}{\int}}\left[{\lambda}^{r1}{\displaystyle \sum _{x=r}^{\infty}x\frac{{e}^{\lambda}{\lambda}^{xr}}{\left(xr\right)!}}\right]}\cdot \frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
Taking $x+r$
in place of $x$
, we get
${\mu}_{\left(r\right)}{}^{\prime}={\displaystyle \underset{0}{\overset{\infty}{\int}}{\lambda}^{r1}\left[{\displaystyle \sum _{x=0}^{\infty}\left(x+r\right)\frac{{e}^{\lambda}{\lambda}^{x}}{x!}}\right]}\cdot \frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
It is obvious that the expression within the bracket is $\lambda +r$
and hence, we have
${\mu}_{\left(r\right)}{}^{\prime}=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}{\displaystyle \underset{0}{\overset{\infty}{\int}}{\lambda}^{r1}\left(\lambda +r\right)}\cdot \left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
Using gamma integral and little algebraic simplification, we get finally, a general expression for the $r$
th factorial moment of the ZTPLD (2.2.1) as
${\mu}_{\left(r\right)}{}^{\prime}=\frac{r!\text{\hspace{0.17em}}{\left(\theta +1\right)}^{2}\left(r+\theta +1\right)}{{\theta}^{r}\text{\hspace{0.17em}}\left({\theta}^{2}+3\theta +1\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}};r=1,2,3,\mathrm{...}$
(3.2.1)
Substituting $r=1,2,3,\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}4$
in (3.2.1), the first four factorial moment can be obtained and then using the relationship between factorial moments and moments about origin, the first four moments about origin of the ZTPLD (2.2.1) are given by
${\mu}_{1}{}^{\prime}=\frac{{\left(\theta +1\right)}^{2}\left(\theta +2\right)}{\theta \left({\theta}^{2}+3\theta +1\right)}$
${\mu}_{2}{}^{\prime}=\frac{{\left(\theta +1\right)}^{2}\left({\theta}^{2}+4\theta +6\right)}{{\theta}^{2}\left({\theta}^{2}+3\theta +1\right)}$
${\mu}_{3}{}^{\prime}=\frac{{\left(\theta +1\right)}^{2}\left({\theta}^{3}+8{\theta}^{2}+24\theta +24\right)}{{\theta}^{3}\left({\theta}^{2}+3\theta +1\right)}$
${\mu}_{4}{}^{\prime}=\frac{{\left(\theta +1\right)}^{2}\left({\theta}^{4}+16{\theta}^{3}+78{\theta}^{2}+168\theta +120\right)}{{\theta}^{4}\left({\theta}^{2}+3\theta +1\right)}$
Generating function: The probability generating function of the ZTPLD (2.2.1) is obtained as
${P}_{X}\left(t\right)=E\left({t}^{X}\right)=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}{\displaystyle \sum _{x=1}^{\infty}{t}^{x}}\frac{x+\theta +2}{{\left(\theta +1\right)}^{x}}$
$=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\left[{\displaystyle \sum _{x=1}^{\infty}x{\left(\frac{t}{\theta +1}\right)}^{x}+\left(\theta +2\right){\displaystyle \sum _{x=1}^{\infty}{\left(\frac{t}{\theta +1}\right)}^{x}}}\right]$
$=\frac{{\theta}^{2}\text{\hspace{0.17em}}t}{{\theta}^{2}+3\theta +1}\left[\frac{\theta +1}{{\left(\theta +1t\right)}^{2}}+\frac{\theta +2}{\theta +1t}\right]$
The moment generating function of the ZTPLD (2.2.1) is thus given by
${M}_{X}\left(t\right)=E\left({e}^{t\text{\hspace{0.17em}}X}\right)=\frac{{\theta}^{2}\text{\hspace{0.17em}}{e}^{t}}{{\theta}^{2}+3\theta +1}\left[\frac{\theta +1}{{\left(\theta +1{e}^{t}\right)}^{2}}+\frac{\theta +2}{\theta +1{e}^{t}}\right]$
A Simple method of finding moments of ZTPLD
Using (2.2.4), the $r$
th moment about origin of ZTPLD (2.2.1) can be obtained as
${{\mu}^{\prime}}_{r}=E\left[E\left({X}^{r}\lambda \right)\right]$
$=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}{\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[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
(4.1)
It is obvious that the expression under the bracket in (4.1) is the $r$
th moment about origin of the SBPD. Taking $r=1$
in (4.1) and using the first moment about origin of the SBPD, the first moment about origin of the ZTPLD (2.2.1) is obtained as
${{\mu}^{\prime}}_{1}=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left(\lambda +1\right)}\left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
$=\frac{{\left(\theta +1\right)}^{2}\left(\theta +2\right)}{\theta \left({\theta}^{2}+3\theta +1\right)}$
(4.2)
Again taking $r=2$
in (4.1) and using the second moment about origin of the SBPD, the second moment about origin of the ZTPLD (2.2.1) is obtained as
${{\mu}^{\prime}}_{2}=\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}{\displaystyle \underset{0}{\overset{\infty}{\int}}\left({\lambda}^{2}+3\lambda +1\right)}\left[\left(\theta +1\right)\lambda +\left(\theta +2\right)\right]{e}^{\theta \text{\hspace{0.17em}}\lambda}d\lambda $
$=\frac{{\left(\theta +1\right)}^{2}\left({\theta}^{2}+4\theta +6\right)}{{\theta}^{2}\left({\theta}^{2}+3\theta +1\right)}$
(4.3)
Similarly, taking $r=3\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}4$
in (4.1) and using the respective moments of SBPD, we get finally, after a little simplification, the third and the fourth moments about origin of the ZTPLD (2.2.1) as
${{\mu}^{\prime}}_{3}=\frac{{\left(\theta +1\right)}^{2}\left({\theta}^{3}+8{\theta}^{2}+24\theta +24\right)}{{\theta}^{3}\left({\theta}^{2}+3\theta +1\right)}$
(4.4)
${{\mu}^{\prime}}_{4}=\frac{{\left(\theta +1\right)}^{2}\left({\theta}^{4}+16{\theta}^{3}+78{\theta}^{2}+168\theta +120\right)}{{\theta}^{4}\left({\theta}^{2}+3\theta +1\right)}$
(4.5)
Estimation of parameter
Estimation of Parameter of ZTPD
Maximum Likelihood Estimate (MLE): 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 MLE $\widehat{\theta}$
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
Method of Moment Estimate (MOME): 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). Equating the first population moment about origin to the corresponding sample moment, the MOME $\tilde{\theta}$
of $\theta $
of ZTPD (2.1.1) is 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
Thus both MLE and MOME give the same estimate of the parameter $\theta $
of ZTPD (2.1.1).
Figure 1: Graph of probability functions of ZTPD and ZTPLD for different values of their parameter. The left hand side graphs are for ZTPD and right hand side graphs are for ZTPLD.
Estimation of Parameter of ZTPLD
Maximum Likelihood Estimate (MLE): Let ${x}_{1},{x}_{2},\mathrm{...},{x}_{n}$
be a random sample of size $n$
from the ZTPLD (2.2.1) and let ${f}_{x}$
be the observed frequency in the sample corresponding to $X=x(x=1,2,3,\mathrm{...},k)$
such that $\sum _{x=1}^{k}{f}_{x}}=n$
, where$k$
is the largest observed value having nonzero frequency. The likelihood function $L$
of the ZTPLD (2.2.1) is given by
$L={\left(\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\right)}^{n}\frac{1}{{\left(\theta +1\right)}^{{\displaystyle \sum _{x=1}^{k}x{f}_{x}}}}{{\displaystyle \prod _{x=1}^{k}\left(x+\theta +2\right)}}^{{f}_{x}}$
(5.1.1)
The log likelihood function is given by
$\mathrm{log}L=n\mathrm{log}\left(\frac{{\theta}^{2}}{{\theta}^{2}+3\theta +1}\right){\displaystyle \sum _{x=1}^{k}x\text{\hspace{0.17em}}{f}_{x}\mathrm{log}\left(\theta +1\right)}+{\displaystyle \sum _{x=1}^{k}{f}_{x}\mathrm{log}\left(x+\theta +2\right)}$
and the log likelihood equation is thus obtained as
$\frac{d\mathrm{log}L}{d\theta}=\frac{2n}{\theta}\frac{n\left(2\theta +3\right)}{{\theta}^{2}+3\theta +1}\frac{n\text{\hspace{0.17em}}\overline{x}}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{{f}_{x}}{x+\theta +2}}$
The maximum likelihood estimate $\widehat{\theta}$
of $\theta $
is the solution of the equation $\frac{d\mathrm{log}L}{d\theta}=0$
and is given by the solution of the following nonlinear equation
$\frac{2n}{\theta}\frac{n\left(2\theta +3\right)}{{\theta}^{2}+3\theta +1}\frac{n\text{\hspace{0.17em}}\overline{x}}{\theta +1}+{\displaystyle \sum _{x=1}^{k}\frac{{f}_{x}}{x+\theta +2}}=0$
(5.1.2)
where $\overline{x}$
is the sample mean. This nonlinear equation can be solved by any numerical iteration methods such as Newton Raphson method, Bisection method, Regula –Falsi method etc. Ghitany et al. [20] showed that the MLE $\widehat{\theta}$
of $\theta $
is consistent and asymptotically normal.
Method of Moment Estimate (MOME): 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 ZTPLD (2.2.1). Equating the first population moment about origin to the corresponding sample moment, the MOME $\widehat{\theta}$
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 $\left(\overline{x}1\right){\theta}^{3}+\left(3\overline{x}4\right){\theta}^{2}+\left(\overline{x}5\right)\theta 2=0\text{\hspace{0.17em}}\text{\hspace{0.17em}};\overline{x}>1$
is the sample mean. Ghitany et al. [20] showed that the MOME $\widehat{\theta}$ of $\theta $
is consistent and asymptotically normal.
Applications
In this section, both ZTPD and ZTPLD have been fitted to a number of datasets using maximum likelihood estimates relating to demography, biological sciences, and social sciences to test their goodness of fits and it has been observed that in most of the cases ZTPLD gives much closer fits than ZTPD and in some cases ZTPD gives much closer fits than ZTPLD.
Mortality
Mortality does not depend only on biological factors; it depends upon the prevailing health conditions, medical facilities, the socioeconomic and cultural factors. In developing and underdeveloped countries, the mortality among infants and children is found much higher than that among youngsters. The high infant mortality has thrown a serious challenge to the medical personnel and is considered as one of the sensitive position of existing medical and health facilities in the population.
Number of Neonatal Deaths 
Observed Number of Mothers 
Expected Frequency 
ZTPD 
ZTPLD 
1 
409 
399.7 
408.1 
2 
88 
102.3 
89.4 
3
4
5 
19
5
1 
$\begin{array}{l}17.5\\ 2.2\\ 0.3\end{array}\}$

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

Total 
522 
522.0 
522.0 
ML Estimate 

$\widehat{\theta}=0.512047$

$\widehat{\theta}=4.199697$

${\chi}^{2}$


3.464 
0.145 
d.f. 

1 
2 
Pvalue 

0.0627 
0.9301 
Table 1: The number of mothers of the rural area having at least one live birth and one neonatal death.
Number of Neonatal Deaths 
Observed Number of Mothers 
Expected Frequency 
ZTPD 
ZTPLD 
1 
71 
66.5 
72.3 
2 
32 
35.1 
28.4 
3
4
5 
7
5
3 
$\begin{array}{l}12.3\\ 3.3\\ 0.8\end{array}\}$

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

Total 
118 
118.0 
118.0 
ML Estimate 

$\widehat{\theta}=1.055102$

$\widehat{\theta}=2.049609$

${\chi}^{2}$ 

0.696 
2.274 
d.f. 

1 
2 
Pvalue 

0.4041 
0.3208 
Table 2: The number of mothers of the estate area having at least one live birth and one neonatal death.
Number of Infant and Child Deaths 
Observed Number of Mothers 
Expected Frequency 
ZTPD 
ZTPLD 
1 
176 
164.3 
171.6 
2 
44 
61.2 
51.3 
3
4
5 
16
6
2 
$\begin{array}{l}15.2\\ 2.8\\ 0.5\end{array}\}$

15.0
$\begin{array}{l}4.3\\ 1.7\end{array}\}$

Total 
244 
244.0 
244.0 
ML Estimate 

$\widehat{\theta}=0.744522$

$\widehat{\theta}=2.209411$

${\chi}^{2}$ 

7.301 
1.882 
d.f. 

1 
2 
Pvalue 

0.0069 
0.3902 
Table 3: The number of mothers of the urban area with at least two live births by the number of infant and child deaths.
Number of Infant and Child Deaths 
Observed Number of Mothers 
Expected Frequency 
ZTPD 
ZTPLD 
1 
745 
708.9 
738.1 
2 
212 
255.1 
214.8 
3 
50 
61.2 
61.3 
4
5
6 
21
7
3 
$\begin{array}{l}11.0\\ 1.6\\ 0.2\end{array}\}$

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

Total 
1038 
1038.0 
1038.0 
ML Estimate 

$\widehat{\theta}=0.719783$

$\widehat{\theta}=3.007722$

${\chi}^{2}$ 

37.046 
4.773 
d.f. 

2 
3 
Pvalue 

0.0 
0.1892 
Table 4: The number of mothers of the rural area with at least two live births by the number of infant and child deaths.
Number of Infant Deaths 
Observed Number of Mothers 
Expected Frequency 
ZTPD 
ZTPLD 
1 
683 
659.0 
674.4 
2 
145 
177.4 
154.1 
3
4
5 
29
11
5 
$\begin{array}{l}31.8\\ 4.3\\ 0.5\end{array}\}$

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

Total 
873 
873.0 
873.0 
ML Estimate 

$\widehat{\theta}=0.538402$

$\widehat{\theta}=4.00231$

${\chi}^{2}$ 

8.718 
5.310 
d.f. 

1 
2 
Pvalue 

0.0031 
0.0703 
Table 5: The number of literate mothers with at least one live birth by the number of infant deaths.
Number of Child Deaths 
Observed Number of Mothers 
Expected Frequency 
ZTPD 
ZTPLD 
1 
89 
76.8 
83.4 
2 
25 
39.9 
32.3 
3
4
5
6 
11
6
3
1 
$\begin{array}{l}13.8\\ 3.6\\ 0.7\\ 0.2\end{array}\}$

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

Total 
135 
135.0 
135.0 
ML Estimate 

$\widehat{\theta}=1.038289$

$\widehat{\theta}=2.089084$

${\chi}^{2}$ 

7.90 
3.428 
d.f. 

1 
2 
Pvalue 

0.0049 
0.1801 
Table 6: The number of mothers of the completed fertility having experienced at least one child death.
Number Of Neonatal Deaths 
Observed Number Of Mothers 
Expected Frequency 
ZTPD 
ZTPLD 
1 
567 
545.8 
561.4 
2 
135 
162.5 
139.7 
3 
28 
32.3 
34.2 
4
5 
11
5 
$\begin{array}{l}4.8\\ 0.6\end{array}\}$

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

Total 
746 
746.0 
746.0 
ML Estimate 

$\widehat{\theta}=0.595415$

$\widehat{\theta}=3.625737$

${\chi}^{2}$ 

26.855 
3.839 
d.f. 

2 
2 
Pvalue 

0.0 
0.1467 
Table 7: The number of mothers having at least one neonatal death.
In this section, an attempt has been made to test the suitability of ZTPD and ZTPLD in describing the neonatal deaths as well as of infant and child deaths experienced by mothers. The datasets considered here are the data of Sri Lanka and India. The datasets of Meegama et al. [21] have been used as the data of Sri Lanka whereas the data from the survey conducted by Lal [22] and the survey of Kadam Kuan, Patna, conducted in 1975 and referred to by Mishra [23] have been used as the data of India. It is obvious from the fittings of ZTPD and the ZTPLD that ZTPLD gives much closer fits in almost all cases except in Table 2. Hence, in case of demographic data, ZTPLD is a better alternative than ZTPD to model count data.
Biological Sciences
In this section, an attempt has been made to test the goodness of fit of both ZTPD and ZTPLD on many data sets relating to biological sciences. It has been observed that ZTPLD gives much closer fits than ZTPD in almost all cases except in Table 11 regarding the distribution of the number of leaf spot grade of Ichinose variety of Mulberry. Thus in biological sciences ZTPLD is a better model than ZTPD to model zerotruncated count data.
Number of European Red Mites 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
38 
28.7 
36.1 
2 
17 
25.7 
20.5 
3 
10 
15.3 
11.2 
4
5
6
7
8 
9
3
2
1
0 
$\begin{array}{l}6.9\\ 2.5\\ 0.7\\ 0.2\\ 0.1\end{array}\}$

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

Total 
80 
80.0 
80.0 
ML Estimate 

$\widehat{\theta}=1.791615$

$\widehat{\theta}=1.185582$

${\chi}^{2}$ 

9.827 
2.467 
d.f. 

2 
3 
Pvalue 

0.0073 
0.4813 
Table 8: Number of European red mites on apple leaves, reported by Garman [24].
Number of Yeast Cells Counts Per Mm Square 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
128 
121.3 
127.6 
2 
37 
49.2 
40.9 
3
4
5
6 
18
3
1
0 
$\begin{array}{l}13.3\\ 2.7\\ 0.4\\ 0.1\end{array}\}$

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

Total 
187 
187.0 
187.0 
ML Estimate 

$\widehat{\theta}=0.811276$

$\widehat{\theta}=2.667323$

${\chi}^{2}$ 

5.228 
1.034 
d.f. 

1 
1 
Pvalue 

0.0222 
0.3092 
Table 9: Number of yeast cell counts observed per mm square, reported by Student [25].
Number of Fly Eggs 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
22 
15.3 
26.8 
2 
18 
21.9 
19.8 
3 
18 
20.8 
13.9 
4 
11 
14.9 
9.5 
5 
9 
8.5 
6.4 
6
7
8
9 
6
3
0
1 
$\begin{array}{l}4.1\\ 1.7\\ 0.6\\ 0.3\end{array}\}$

$\begin{array}{l}4.2\\ 2.7\\ 1.7\\ 3.0\end{array}\}$

Total 
88 
88.0 
88.0 
ML Estimate 

$\widehat{\theta}=2.860402$

$\widehat{\theta}=0.718559$

${\chi}^{2}$ 

6.677 
3.743 
d.f. 

4 
4 
Pvalue 

0.1540 
0.4419 
Table 10: The number of counts of flower heads as per the number of fly eggs reported by Finney & Varley [26].
Number Of Leaf Spot Grade 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
18 
14.2 
23.0 
2 
15 
18.7 
16.3 
3 
10 
16.5 
11.1 
4 
14 
10.9 
7.3 
5 
13 
9.7 
12.4 
Total 
70 
70.0 
70.0 
ML Estimate 

$\widehat{\theta}=2.639984$

$\widehat{\theta}=0.781902$

${\chi}^{2}$ 

6.311 
7.476 
d.f. 

3 
3 
Pvalue 

0.0974 
0.0582 
Table 11: The number of leaf spot grade of Ichinose variety of Mulberry, reported by Khurshid [27].
Number of Leaf Spot Grade 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
37 
28.5 
36.7 
2 
16 
26.7 
21.4 
3 
15 
16.7 
12.0 
4
5 
8
8 
$\begin{array}{l}7.8\\ 4.2\end{array}\}$

6.6
7.3 
Total 
84 
84.0 
84.0 
ML Estimate 

$\widehat{\theta}=1.874567$

$\widehat{\theta}=1.130211$

${\chi}^{2}$ 

8.329 
2.477 
d.f. 

2 
3 
Pvalue 

0.0155 
0.4795 
Table 12: The number of leaf spot grade of Kokuso20 variety of Mulberry, reported by Khurshid [27].
Number of Sites With Particles 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
122 
115.9 
124.8 
2 
50 
57.4 
46.8 
3 
18 
18.9 
17.1 
4
5 
4
4 
$\begin{array}{l}4.7\\ 1.1\end{array}\}$

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

Total 
198 
198.0 
198.0 
ML Estimate 

$\widehat{\theta}=0.990586$

$\widehat{\theta}=2.18307$

${\chi}^{2}$ 

2.14 
0.51 
d.f. 

2 
2 
Pvalue 

0.3430 
0.7749 
Table 13: The number of counts of sites with particles from Immuno gold data, reported by Mathews [28].
Number of Times Hares Caught 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
184 
176.6 
182.6 
2 
55 
66.0 
55.3 
3
4
5 
14
4
4 
$\begin{array}{l}16.6\\ 3.1\\ 0.7\end{array}\}$

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

Total 
261 
261.0 
261.0 
ML Estimate 

$\widehat{\theta}=0.756171$

$\widehat{\theta}=2.863957$

${\chi}^{2}$ 

2.45 
0.61 
d.f. 

1 
2 
Pvalue 

0.1175 
0.7371 
Table 14: The number of snowshoe hares counts captured over 7 days, reported by Keith & Meslow [29].
Social Sciences
In this section, an attempt has been made to test the goodness of fit test of both ZTPD and ZTPLD on many datasets relating to social sciences, such as migration, number of accidents and freeforming small group size. It has been observed that the ZTPD gives much closer fits than ZTPLD in almost all cases except the distribution of the number of household having at least one migrant in Table 15.
X 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
375 
354.0 
379.0 
2 
143 
167.7 
137.2 
3 
49 
53.0 
48.4 
4
5
6
7
8 
17
2
2
1
1 
$\begin{array}{l}12.5\\ 2.4\\ 0.4\\ 0.1\\ 0.0\end{array}\}$

16.8
$\begin{array}{l}5.7\\ 1.9\\ 0.6\\ 0.3\end{array}\}$

Total 
590 
590.0 
590.0 
ML Estimate 

$\widehat{\theta}=0.947486$

$\widehat{\theta}=2.284782$

${\chi}^{2}$ 

8.933 
1.031 
d.f. 

2 
3 
Pvalue 

0.0115 
0.7937 
Table 15: Number of households having at least one migrant according to the number of migrants, reported by Sing & Yadav [30].
Number of Accidents 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
2039 
2034.2 
2050.4 
2 
312 
319.5 
291.7 
3
4
5 
35
3
1 
33.5
2.6
0.2 
41.1
5.8
1.0 
Total 
2390 
2390.0 
2390.0 
ML Estimate 

$\widehat{\theta}=0.314125$

$\widehat{\theta}=6.749732$

${\chi}^{2}$ 

0.387 
3.128 
d.f. 

1 
1 
Pvalue 

0.5339 
0.0769 
Table 16: Number of workers according to the number of accidents, reported by Mir & Ahmad [31].
Number of Pairs of Running Shoes 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
18 
17.7 
24.1 
2 
18 
18.5 
15.0 
3 
12 
12.9 
9.0 
4
5 
7
5 
$\begin{array}{l}6.7\\ 4.2\end{array}\}$

5.2
6.2 
Total 
60 
60.0 
60.0 
ML Estimate 

$\widehat{\theta}=2.087937$

$\widehat{\theta}=1.004473$

${\chi}^{2}$ 

0.191 
3.998 
d.f. 

2 
3 
Pvalue 

0.9089 
0.2617 
Table 17: Number of counts of pairs of running shoes owned by 60 members of an athletics club, reported by Simonoff [32].
Group Size 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
1486 
1500.5 
1592.8 
2 
694 
669.6 
551.8 
3 
195 
199.2 
186.5 
4 
37 
44.4 
61.9 
5
6 
10
1 
$\begin{array}{l}7.9\\ 1.3\end{array}\}$

$\begin{array}{l}20.3\\ 9.6\end{array}\}$

Total 
2423 
2423.0 
2423.0 
ML Estimate 

$\widehat{\theta}=0.892496$

$\widehat{\theta}=2.419103$



2.702 
66.155 
d.f. 

3 
3 
Pvalue 

0.4399 
0.0 
Table 18: Number of free forming small group size according to the group size, reported by Coleman & James [33].
Group Size 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
316 
316.4 
335.8 
2 
141 
140.7 
116.0 
3 
44 
41.7 
39.1 
4
5 
5
4 
$\begin{array}{l}9.3\\ 1.9\end{array}\}$

$\begin{array}{l}12.9\\ 6.2\end{array}\}$

Total 
510 
510.0 
510.0 
ML Estimate 

$\widehat{\theta}=0.889458$

$\widehat{\theta}=2.428125$

${\chi}^{2}$ 

0.558 
12.481 
d.f. 

2 
2 
Pvalue 

0.7565 
0.0019 
Table 19: Number of free forming small group size according to the group size, reported by Coleman & James [33].
Group Size 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
306 
302.5 
322.5 
2 
132 
139.5 
114.6 
3 
47 
42.9 
39.7 
4
5 
10
2 
$\begin{array}{l}9.9\\ 2.1\end{array}\}$

$\begin{array}{l}13.5\\ 6.8\end{array}\}$

Total 
497 
497.0 
497.0 
ML Estimate 

$\widehat{\theta}=0.922509$

$\widehat{\theta}=2.341269$

${\chi}^{2}$ 

0.834 
8.220 
d.f. 

2 
2 
Pvalue 

0.6590 
0.0164 
Table 20: Number of free forming small group size according to the group size, reported by Coleman & James [33].
Group Size 
Observed Frequency 
Expected Frequency 
ZTPD 
ZTPLD 
1 
305 
307.2 
327.7 
2 
144 
142.9 
117.3 
3 
50 
44.3 
40.9 
4
5
6 
5
2
1 
$\begin{array}{l}10.3\\ 1.9\\ 0.3\end{array}\}$

$\begin{array}{l}14.0\\ 4.7\\ 2.4\end{array}\}$

Total 
507 
507.0 
507.0 
ML Estimate 

$\widehat{\theta}=0.930664$

$\widehat{\theta}=2.31943$

${\chi}^{2}$ 

2.376 
17.806 
d.f. 

2 
2 
Pvalue 

0.3048 
0.0001 
Table 21: Number of free forming small group size according to the group size, reported by Coleman & James [33].
Conclusion
In this paper, the nature and behavior of ZTPD and ZTPLD have been studied by drawing different graphs for the different values of its parameter. A general expression for the th factorial moment has been given and the first four moments about origin has been obtained. Also a very simple and easy method for finding moments of ZTPLD has been suggested. An attempt has been made to study the goodness of fit of both ZTPD and ZTPLD to count data relating to demography, biological sciences, and social sciences and it has been found that ZTPLD is a better model than the ZTPD in almost all datasets relating to mortality and biological sciences whereas ZTPD is a better model than ZTPLD in almost all datasets relating to social sciences.. Thus, ZTPLD has an advantage over ZTPD for modeling zerotruncated count data in mortality and biological sciences whereas ZTPD has an advantage over ZTPLD for modeling zertruncated count data in social sciences.
References
 Sankaran M (1970) The discrete PoissonLindley distribution. Biometrics 26: 145149.
 Shanker R and 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: 102107.
 Ghitany ME, Atieh B and Nadarajah S (2008a) Lindley distribution and Its Applications. Mathematics and Computers in Simulation 78(4): 493506.
 Ghitany ME and AlMutairi DK (2009) Estimation Methods for the discrete PoissonLindley distribution. Journal of Statistical Computation and Simulation 79(1): 19.
 Shanker R and Mishra A (2014) A twoparameter PoissonLindley distribution. International Journal of Statistics and Systems 9(1): 7985.
 Shanker R and Mishra A (2013 a) A twoparameter Lindley distribution. Statistics in Transition new Series 14(1): 4556.
 Shanker R and Mishra A (2015) A quasi PoissonLindley distribution (submitted).
 Shanker R and Mishra A (2013 b) A quasi Lindley distribution. African journal of Mathematics and Computer Science Research 6(4): 6471.
 Shanker R, Sharma S and Shanker R (2012) A Discrete twoParameter Poisson Lindley Distribution. Journal of Ethiopian Statistical Association 21:1522.
 Shanker R, Sharma S and Shanker R (2013) A twoparameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics 4: 363368.
 Shanker R and Tekie AL (2014) A new quasi PoissonLindley distribution. International Journal of Statistics and Systems 9(1): 8794.
 Shanker R and Amanuel AG (2013) A new quasi Lindley distribution. International Journal of Statistics and Systems8(2): 143156.
 Shanker R, Hagos F and Sujatha S (2015) On modeling of Lifetimes data using exponential and Lindley distributions. Submitted to Biometrics and Biostatistics International Journal 2(5):17.
 Plackett RL (1953) The truncated Poisson distribution. Biometrics 9: 185188.
 David FN and Johnson NL (1952) The truncated Poisson. Biometrics 8: 275285.
 Cohen AC (1960 a) An extension of a truncated Poisson distribution. Biometrics 16: 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 and Goen RL (1958) MVUE for the truncated Poisson distribution. Annals of Mathematical Statistics 29: 755765.
 Ghitany ME, AlMutairi DK and Nadarajah S (2008b) Zerotruncated PoissonLindley distribution and its Applications. Mathematics and Computers in Simulation 79(3): 279287.
 Meegama SA (1980) Socioeconomic determinants of infant and child mortality in Sri Lanka, an analysis of postwar experience. International Statistical Institute (World Fertility Survey), Netherland.
 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, Unpublished Ph.D thesis, Patna University, Patna, India.
 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.
 Finney DJ and Varley GC (1955) An example of the truncated Poisson distribution. Biometrics 11: 387394.
 Khursid AM (2008) On sizebiased Poisson distribution and its use in zero truncated case. J KSIAM 12(3): 153160.
 Matthews JN and Appleton DR (1993) An application of the truncated Poisson distribution to Immunogold assay. Biometrics 49(2): 617621.
 Keith LB and Meslow EC (1968) Trap response by snowshoe hares. Journal of Wildlife Management 32: 795801.
 Singh SN and Yadava KN (1981) Trends in rural outmigration at household level. Rural Demogr 8(1): 5361.
 Mir KA and Ahmad M (2009) Sizebiased distributions and their applications. Pak J Statist 25(3): 283294.
 Simmonoff JS (2003) Analyzing Categorical data, Springer, New York.
 Coleman JS and James J (1961) The equilibrium size distribution of freelyforming groups. Sociometry 24(1): 3645.

