1. Introduction

Suppose is the original distribution. Then the zero-truncated version of is defined as

(1.1)

In probability theory, zero-truncated distribution is a certain class of discrete distribution whose support is the set of positive integers. When the data to be modeled originate from a mechanism which generates data that structurally excludes zero counts, zero-truncated distribution is the appropriate choice.

The probability mass function (pmf) of Poisson-Lindley distribution (PLD) given by

(1.2)

has been introduced by Sankaran (1970) to model count data. It is a Poisson mixture of Lindley (1958) distribution having probability density function (pdf)

(1.3)

Ghitany et al (2008 a) has detailed study on various properties, estimation of parameter and application of Lindley distribution. Ghitany and Al-Mutairi (2009) have discussed the estimation methods of PLD along with simulation study and application. Shanker et al (2015) has detailed discussion on the applications of exponential and Lindley distributions for modeling lifetimes data from different fields of knowledge. Shanker and Hagos (2015) have discussed the applications of Poisson-Lindley distribution in biological sciences.

Using (1.1) and (1.2), Ghitany et al (2008 b) obtained zero-truncated Poisson -Lindley distribution (ZTPLD) defined by its pmf

(1.4)

Shanker et al (2015) has done comparative study on applications of ZTPLD and zero-truncated Poisson distribution (ZTPD) on different real data sets from different fields of knowledge and showed that ZTPLD gives better fit than ZTPD in almost all data sets relating to demography, biological sciences and social sciences.

The zero-truncated Poisson distribution (ZTPD) is defined by its pmf

(1.5)

In this paper, a zero-truncated Poisson-Shanker distribution (ZTPSD) has been introduced by taking the zero-truncated version of Poisson-Shanker distribution (PSD) suggested by Shanker (2016). The first four moments about origin and the moments about mean of ZTPSD have been obtained and thus expressions for coefficient of variation, skewness, kurtosis, and index of dispersion have been given. The estimation of its parameter has been discussed using maximum likelihood estimation and method of moments. Finally, applications of ZTPSD to three observed real data sets have been given to test its goodness of fit over zero-truncated Poisson distribution (ZTPD) and Zero-truncated Poisson-Lindley distribution (ZTPLD).

2. Zero-Truncated Poisson-Shanker Distribution (ZTPSD)

The Poisson-Shanker distribution defined by its pmf

(2.1)

has been introduced by Shanker (2016) to model count data. The distribution arises from the Poisson distribution when its parameter follows Shanker distribution introduced by Shanker (2015) with pdf

(2.2)

Shanker (2015) showed that (2.2) is a better model than both exponential and Lindley (1958) distributions for modeling lifetime data from biomedical science and engineering.

Using (1.1) and (2.1), the pmf of zero-truncated Poisson-Shanker distribution (ZTPSD) can be obtained as

(2.3)

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

(2.4)

when its parameter follows a distribution having p.d.f.

(2.5)

Thus the p.m.f. of ZTPSD can be obtained as

(2.6)

which is the p.m.f. of ZTPSD, as obtained earlier in (2.3). The main motivation of considering a continuous distribution in (2.5) is that moments of ZTPSD (2.3) can easily be obtained from the size-biased Poisson mixture of the continuous distribution using (2.6).

To have a comparative study on the nature and behavior of ZTPSD and ZTPLD, several graphs of their pmf’s have been drawn for varying values of their parameter and presented in figure 1.

Figure 1. Graphs of ZTPSD and ZTPLD for varying values of the parameter

A separate graph of ZTPSD for varying values of parameter θ has been shown in figure 2.

Figure 2. Graphs of ZTPSD for varying values of the parameter

Since is a decreasing function of is log-concave. Therefore, ZTPSD is unimodal, has increasing failure rate (IFR), and hence increasing failure rate average (IFRA). It is new better than used (NBU), new better than used in expectation (NBUE), and has decreasing mean residual life (DMRL). Detailed discussions about the definitions of these aging concepts are available in Barlow and Proschan (1981).

3. Moments and Related Measures

From (2.6), the rth factorial moment about origin of ZTPSD (2.3) can be obtained as

where

Taking in place of we get

(3.1)

Taking r = 1,2,3 and 4 in (3.1), first four factorial moments about origin can be obtained and then using the relationship between factorial moments and moments about origin, the first four moments about origin of ZTPSD can be obtained as

Using the relationship between moments about origin and moments about mean, the moments about mean of ZTPSD are thus obtained as

The coefficient of variation (C.V), coefficient of Skewness coefficient of Kurtosis and index of dispersion of ZTPSD are thus obtained as

It can be easily verified that the ZTPSD is over dispersed equi-dispersed and under dispersed for respectively. Note that the ZTPLD is over dispersed , equi-dispersed and under dispersed for respectively.

The nature of coefficient of variation, coefficient of skewness, coefficient of kurtosis, and index of dispersion of ZTPSD for varying values of the parameter θ are shown in figure 3.

Figure 3. Graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis, and index of dispersion of ZTPSD for varying values of the parameter θ

To study the comparative nature of mean, variance, coefficient of variation, coefficient of skewness, coefficient of kurtosis, and index of dispersion of ZTPSD and ZTPLD for varying values of the parameter θ, a table for their values have been prepared and presented in table 1.

Table 1. Numerical values of       and       of ZTPSD and ZTPLD

4. Estimation of Parameter

4.1. Maximum Likelihood Estimate (MLE) of Parameter: Let be a random sample of size from the ZTPSD (2.3) and let be the observed frequency in the sample corresponding to such that where is the largest observed value having non-zero frequency. The likelihood function of the ZTPSD (2.3) is given by

The log likelihood function is given by

and the log likelihood equation is thus obtained as

The maximum likelihood estimate of is the solution of the equation and is given by the solution of the following non-linear equation

where 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. In the following theorem, the consistency and asymptotic normality of maximum likelihood estimator of ZTPSD has been established.

Theorem: The ML estimator of of the ZTPSD is consistent and asymptotically normal. That is where

is the Fisher’s information about

Proof: The ZTPSD satisfies the regularity conditions under which the ML estimator of is consistent and asymptotically normal [see Hogg et al (2005), chapter 6]. We have

(4.1.3)

where

(4.1.4)

(4.1.5)

and

(4.1.6)

Using equations (4.1.4), (4.1.5), and (4.1.6) in (4.1.3), we get

4.2. Method of Moment Estimate (MOME) of Parameter: Equating the population mean to the corresponding sample mean, MOME of of ZTPSD is the solution of the following non-linear equation

where is the sample mean.

5. Applications to Real Data Sets

The ZTPSD has been fitted to a number of data - sets to test its goodness of fit over ZTPD and ZTPLD. The maximum likelihood estimate (MLE) has been used to fit the ZTPSD, ZTPLD and ZTPD. Three examples of observed data-sets, for which the ZTPD, ZTPLD and ZTPSD has been fitted, are presented. The first data-set is the number of flower heads as per the number of fly eggs reported by Finney and Varley (1955), the second data- set is the number of yeast cell counts observed per mm square reported by Student (1907) and the third data-set is animal abundance data of Keith and Meslow (1968) regarding the distribution of snowshoe hares captured over 7 days.

Table 2. The numbers of counts of flower heads as per the number of fly eggs reported by Finney and Varley (1955)

Table 3. Number of yeast cell counts observed per mm square reported by Student (1907)

Table 4. Number of Snowshoe hares counts captured over 7 days reported by Keith and Meslow (1968)

6. Conclusions

A zero-truncated Poisson-Shanker distribution (ZTPSD) has been introduced. The first four moments about origin and the moments about mean have been obtained. The expressions for coefficient of variation, skewness, kurtosis and index of dispersion of ZTPSD have been given. The method of maximum likelihood and the method of moments have also been discussed for estimating its parameter. Three examples of observed real data- sets have been given to test its goodness of fit over ZTPD and ZTPLD. The goodness of fit of ZTPSD gives quite satisfactory fit over both ZTPD and ZTPLD and thus ZTPSD can be considered an important distribution for modeling zero-truncated count data.

ACKNOWLEDGEMENTS

The author is grateful to the editor in-chief and the anonymous reviewer for their constructive comments which improved the presentation of the paper.