The Discrete Poisson-Shanker Distribution
The Shanker distribution defined by its probability density function (p.d.f.) and corresponding cumulative distribution function (c.d.f)
has been introduced by Shanker  for modeling real lifetime data-set from engineering and biomedical science. It has been shown by Shanker  that the p.d.f. (1.1) is a two-component mixture of an exponential distribution with scale parameter
ϴ and a gamma distribution having shape parameter 2 and a scale parameter ϴ with their mixing proportions and respectively. Shanker  has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability , amongst others.
A number of discrete distributions have been introduced in Statistics and the main reason for so many discrete distributions is that each distribution is based on certain assumptions and has specific applications due to its shape and distributional properties. It has been observed that a particular distribution will not fit well on all discrete data due to various reasons including the variation in the data, shape of the distribution, assumptions of the distribution, some among others. Therefore, a search for new discrete distribution is still going on which can fit data well as compared to existing distributions. Shanker distribution, as shown by Shanker , is a better model than the Lindley and exponential distributions for modeling lifetime data from biomedical science and engineering, it is expected that the Poisson mixture of Shanker distribution named Poisson-Shanker distribution (PSD) will provide a better fit than the Poisson-Lindley distribution (PLD) [2,5], a Poisson mixture of Lindley distribution [3,4], for modeling count data. Further, since both PLD and PSD are of one parameter, a comparative study between PLD and PSD is justifiable from fitting discrete data. The main objective of the paper is to firstly find the Poisson mixture of Shanker distribution and have detailed study about its distributional properties, estimation of parameter and applications over some discrete distributions.
In the present paper, a Poisson mixture of Shanker distribution introduced by Shanker  named, “Poisson-Shanker distribution (PSD)” has been proposed. Its various mathematical and statistical properties including its shape, moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate, over-dispersion, unimodality etc have been discussed. The estimation of its parameter has been discussed using maximum likelihood estimation and method of moments. The goodness of fit of PSD along with Poisson distribution and Poisson-Lindley distribution (PLD), a Poisson mixture of Lindley  distribution and introduced by Sankaran , have been discussed.
Suppose the parameter λ of the Poisson distribution follows Shanker distribution . Then the Poisson mixture of Shanker distribution can be obtained as
Figure 1. Graphs of probability mass function of PSD and PLD for different values of the parameter ϴ
Moments and Related Measures
Substituting r= 1,2,3,and 4 in (3.1), the first four factorial moments about origin can be obtained and using the relationship between factorial moments about origin and moments about origin, the first four moment about origin of the PSD (2.2) are obtained as
Method of Moment Estimate (MOME): Let X1,X2,…… Xn be
a random sample of size n from the PSD (2.2). Equating the
population mean to the corresponding sample mean, the
MOME of ϴ is the solution of the following cubic equation
Applications and Goodness of Fit
The PSD has been fitted to a number of data – sets to test its goodness of fit over Poisson distribution (PD) and Poisson- Lindley distribution (PLD). The maximum likelihood estimate (MLE) has been used to fit the PSD. Four examples of observed data-sets, for which the PD, PLD and PSD has been fitted, are presented. The data-set in table 2 is due to Kemp and kemp  regarding the distribution of mistakes in copying groups of random digits, the data-set in table 3 is due to Beall  regarding the distribution of Pyrausta nublilalis, the data-set in table 4 is the number of accidents to 647 women working on high explosive shells in 5 weeks available in Sankaran , and the data-set in table in 5 is distribution of number of Chromatid aberrations, available in Loeschke and Kohler  and Janardan and Schaeffer . The fitting of PSD shows that it is a better model than both Poisson and Poisson –Lindley distributions and hence it has advantage over both Poisson and Poisson-Lindley distributions. The PSD can be considered to be an important discrete distribution because of its flexibility over Poisson and Poisson-Lindley distributions.
where is the sample mean.
In this paper “Poisson-Shanker distribution (PSD)” has been obtained by compounding Poisson distribution with Shanker distribution introduced by Shanker . The expression for the r th factorial moment has been derived and hence the first four moments about origin and the moments about mean has been given. The expression for coefficient of variation, skewness, kurtosis, and index of dispersion has been obtained. Some of its important mathematical and statistical properties have been discussed. The maximum likelihood estimation and the method of moments for estimating its parameter have been discussed. The distribution has been fitted using maximum likelihood estimate to some data- sets to test its goodness of fit over Poisson distribution (PD) and Poisson-Lindley distribution (PLD) and observed that PSD gives much closer fit than PD and PLD in all data-sets.
The author expresses his thankfulness to both editor-in-chief and the reviewer for valuable comments which improved the quality and the presentation of the paper.
10. Loeschke V, Kohler W. Deterministic and Stochastic models of the negative binomial distribution and the analysis of chromosomal aberrations in human leukocytes. Biometrical Journal. 1976, 18(6): 427-451.