1. Introduction

Shanker (2016) introduced Poisson-Shanker distribution (PSD) having probability mass function (p.m.f.)

(1.1)

for modeling count data from different fields of knowledge. The PSD arises from the Poisson distribution when its parameter follows a lifetime distribution named Shanker distribution introduced by Shanker (2015) having probability density function (p.d.f.)

(1.2)

Size-biased distributions are special class of weighted distributions and arise in practice when observations from a sample are recorded with probability proportional to some measure of unit size. Fisher (1934) firstly introduced these distributions to model ascertainment biases which were later formalized by Rao (1965) in a unifying theory. Size-biased observations occur in many research areas and its fields of applications includes econometrics, environmental science, medical science, sociology, psychology, ecology, geological sciences etc. The applications of size-biased distribution theory in fitting distributions of diameter at breast height (DBH) data arising from horizontal point sampling (HPS) has been discussed by Van Deusen (1986). Further, Lappi and Bailey (1987) have studied and applied size-biased distributions to analyze HPS diameter increment data. The detailed statistical applications of size-biased distributions to the analysis of data relating to human population and ecology can be found in Patil and Rao (1977, 1978). A number of research have been done relating to size-biased distributions and their applications in different areas of knowledge by many researchers including Scheaffer (1972), Patil and ord (1976), Singh and Maddala (1976), Patil, (1981), McDonald (1984), Gove (2000, 2003), Correa and Wolfson (2007), Drummer and McDonald (1987), Ducey (2009), Alavi and Chinipardaz (2009), Ducey and Gove (2015), are some among others.

Let a random variable has original probability distribution Suppose the sample units are weighted or selected with probability proportional to some measure where is a positive integer. Then the corresponding size-biased probability distribution of order can be defined by its probability mass function

where If then the distribution is known as simple size-biased and is applicable for size-biased sampling in sampling theory. If , then the distribution is known as area-biased distribution and is applicable for area-biased sampling in forestry.

The p.m.f. of the size-biased Poisson-Shanker distribution (SBPSD) with parameter can thus be be obtained as

(1.3)

where is the mean of the Poisson-Shanker distribution (PSD) with p.m.f. (1.1).

Recall that the probability mass function of size-biased Poisson-Lindley distribution (SBPLD) with parameter obtained by Ghitany and Mutairi (2008), is given by

(1.4)

The SBPLD is a size-biased version of Poisson-Lindley distribution introduced by Sankaran (1970) and the Poisson-Lindley distribution is a Poisson mixture of Lindley distribution, introduced by Lindley (1958). The statistical and mathematical properties, estimation of parameter using both maximum likelihood estimation and method of moments, and goodness of fit of SBPLD have been discussed by Ghitany and Mutairi (2008). Shanker et al (2015) has discussed the applications of SBPLD for modeling data on thunderstorms and found that SBPLD is a better model for thunderstorms than size-biased Poisson distribution (SBPD).

The comparative nature and behavior of SBPSD and SBPLD are shown in the figure 1.

Figure 1. Graphs of SBPSD and SBPLD for varying values of parameter

The independent graphs of SBPSD for varying values of the parameter θ have been shown in figure 2.

Figure 2. Graphs of SPBSD for varying values of parameter

The probability mass function of SBPSD can also be obtained from the size-biased Poisson distribution (SPBD) with p.m.f.

(1.5)

when its parameter follows size-biased Shanker distribution (SBSD) with p.d.f.

(1.6)

Thus, the pmf of SBPSD can be obtained as

(1.7)

which is the p.m.f of SBPSD as given in (1.3).

2. Moments and Related Measures of SBPSD

Using (1.7), th rth factorial moment about origin of the SBPSD (1.3) can be obtained as

Taking in place of we get

(2.1)

Substituting and 4 in (2.1), first four factorial moments about origin can be obtained and then using the relationship between factorial moments about origin and moments about origin, the first four moments about origin of SBPSD (1.3) can be obtained as

Again using the relationship between moments about mean and the moments about origin, the moments about mean of the SBPSD (1.3) are obtained as

The coefficient of variation coefficient of Skewness coefficient of Kurtosis and index of dispersion of the SBPSD (1.3) are thus obtained as

It can be easily verified that SBPSD is over-dispersed equi-dispersed and under-dispersed for The graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis, and index of dispersion of SBPSD for varying values of parameter are shown in figure 3.

Figure 3. Graphs of C.V, of the SBPSD for varying values of parameter

To study the comparative nature and behavior of SBPSD and SBPLD, a table for and has been prepared for varying values of the parameter and presented in table 1.

Table 1. Values of       of SBPSD and SBPLD for varying values of the parameter θ

The comparative graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis, and index of dispersion of SBPSD and SBPLD are shown in figure 4.

Figure 4. Graphs of C.V, of the SBPSD and SBPLD for varying values of parameter

3. Statistical Properties of SBPSD

3.1. Reliability Properties

Since

is a deceasing function of is log-concave. Therefore, SBPSD is unimodal, has an increasing failure rate (IFR), and hence increasing failure rate average (IFRA). It is new better than used in expectation (NBUE) and has decreasing mean residual life (DMRL). Detailed discussion and definitions of these aging concepts can be seen in Barlow and Proschan (1981).

3.2. Generating Function

Probability Generating Function: The probability generating function of the SBPSD (1.3) can be obtained as

Moment Generating Function: The moment generating function of the SBPSD (1.3) is given by

4. Estimation of Parameter

4.1. Maximum Likelihood Estimate (MLE): Let be a random sample of size n from the SBPSD (1.3) 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 SBPSD (1.3) is given by

The log likelihood function is obtained as

The first derivative of the log likelihood function is given by

where is the sample mean.

The maximum likelihood estimate (MLE), of is the solution of the equation and is given by the solution of the non-linear equation

(4.1.1)

This non-linear equation can be solved by any numerical iteration methods such as Newton- Raphson method, Bisection method, Regula –Falsi method etc. The consistency and asymptotic normality of maximum likelihood estimator of SBPSD has been established in the following theorem.

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

is the Fisher’s information about

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

(4.1.2)

where

(4.1.3)

Now

(4.1.4)

and

(4.1.5)

Using equations (4.1.3), (4.1.4), and (4.1.5) in (4.1.2), we get

4.2. Method of Moment Estimate (MOME): Let be a random sample of size n from the SBPSD (1.3). Equating population mean to the corresponding sample mean, the MOME of of SBPSD is the solution of the following cubic equation in

(4.2.1)

where is the sample mean.

5. Goodness of Fit of SBPSD

The SBPSD has been fitted to a number of data - sets to test its goodness of fit over SBPD and SBPLD. The maximum likelihood estimate (MLE) has been used to fit the SBPSD. Two examples of observed data-sets, for which the SBPD, SBPLD and SBPSD has been fitted, are presented. The first data-set is immunogold assay data of Cullen et al. (1990) regarding the distribution of number of counts of sites with particles from immunogold assay data and the second data-set is animal abundance data of Keith and Meslow (1968) regarding the distribution of snowshoe hares captured over 7 days.

Table 2. Distribution of number of counts of sites with particles from Immunogold data

Table 3. Distribution of snowshoe hares captured over 7 days

6. Concluding Remarks

In this paper, a size-biased Poisson-Shanker distribution (SBPSD) has been proposed by size-biasing the Poisson-Shanker distribution (PSD) introduced by Shanker (2016). Its raw moments and central moments have been obtained and hence expressions for coefficient of variation (C.V.), skewness, kurtosis and index of dispersion have also been given. Its reliability properties have been explained and generating functions have been derived. The method of maximum likelihood and the method of moments have been discussed for estimating the parameter. Two real lifetime data sets have been presented to test the goodness of fit of SBPSD and the fit is quite satisfactory over size-biased Poisson distribution (SBPD) and size-biased Poisson-Lindley distribution (SBPLD).

ACKNOWLEDGEMENTS

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