1. Introduction

The size - biased distributions arise when the observations generated from a random process do not have equal probability of being recorded and are recorded according to some weight function. When the sampling mechanism is such that the sample units are selected with probability proportional to some measure of the unit size, the resulting distribution is called ‘size-biased distribution’. Fisher (1934) first introduced such distributions to model ascertainment bias and Rao (1965) formulated these in a unifying theory. Patil and Ord (1975) studied the size-biased sampling and the related form-invariant weighted distribution whereas Van Deusen (1986) arrived at size - biased distribution theory independently and applied it to fitting distributions of diameter at breast height (DBH) data arising from horizontal point sampling (HPS). Later, Lappi and Bailey (1987) analyzed HPS diameter increment data using size- biased distribution. Patil and Rao (1977, 1978) examined some general models leading to size - biased distributions. The results were applied to the analysis of data relating to human populations and wild life management. Gove (2003) reviewed some of the recent results on size- biased distributions pertaining to parameter estimation in forestry with special emphasis on Weibull distribution. Simoj and Maya (2006) introduced some fundamental relationships between weighted and unique variables in the context of maintainability function and inverted repair rate. Mir and Ahmad (2009), Das and Roy (2011) and Ducey and Gove (2015) have also studied the various aspects of size - biased distributions.

A simple size-biased version of a distribution is given by its probability function where is the mean of the distribution.

Ghitany and Al Mutairi (2008) obtained a size-biased Poisson-Lindley distribution (SBPLD) given by its probability mass function (p.m.f.)

(1.1)

by size biasing the Poisson -Lindley distribution of Sankaran (1970) having pmf

(1.2)

It is to be mentioned that Sankaran (1970) obtained the distribution (1.2) by mixing the Poisson distribution with the Lindley (1958) distribution having pdf

(1.3)

The first four moments about origin of the SBPLD (1.1) have been obtained as

(1.4)

(1.5)

(1.6)

(1.7)

and so its variance as

(1.8)

Shanker and Mishra (2014) obtained a two-parameter Poisson-Lindley distribution (TPPLD) given by its pmf

(1.9)

having its mean,

(1.10)

It can be seen that the PLD (1.2) is a particular case of it at . Shanker and Mishra (2014) have shown that (1.9) is a better model than the PLD of Sankaran (1970) for analyzing different types of count data. This distribution arises from the Poisson distribution when its parameter follows the Shanker and Mishra (2013) two parameter Lindley distribution having probability density function (p.d.f)

(1.11)

In this paper, a size -biased two parameter Poisson-Lindley distribution (SBTPPLD), of which the SBPLD (1.1) is a particular case, has been obtained. A general expression for its rth factorial moment about origin has been obtained and hence its first four moments about origin and central moments are obtained. The expressions for coefficients of variation, skewness, kurtosis and index of dispersion have also been given. The method of maximum likelihood and the method of moments for the estimation of its parameters have been discussed. The distribution has been fitted to some data sets to show that it provides closer fit than the size-biased Poisson distribution (SBPD) and SBPLD. This makes one believe that SBTPPLD is more flexible than the SBPD and SBPLD for analyzing different count data.

2. A Size- Biased Two Parameter Poisson-Lindley Distribution

A size -biased version of the two parameter Poisson-Lindley distribution (SBTPPLD) with parameters and can be obtained as

(2.1)

Taking from (1.9) and from (1.10), we get

(2.2)

It can be easily seen that at , SBTPPLD (2.2) reduces to SBPLD (1.1). The SBTPPLD (2.2) can also be obtained from the size- biased Poisson distribution when its parameter follows a size-biased two parameter Lindley distribution of Shanker and Mishra (2013) with p.d.f.

(2.3)

We have

(2.4)

(2.5)

which is the SBTPPLD, as obtained in (2.2).

Since

(2.6)

is a decreasing function of , is log-concave. This implies that the SBTPPLD is unimodal, has an increasing failure rate (IFR) and so 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). Details about the definitions and relationship of these aging concepts can be seen in Barlow and Proschan (1981).

3. Moments and Related Measures

The rth factorial moment about origin of the SBTPPLD (2.2) can be obtained as

From (2.4), we get

Taking in place of , we get

Clearly the expression within the bracket is and hence we have

Using gamma integral and a little algebraic simplification, a general expression for the rth factorial moment about origin of SBTPPLD is obtained as

(3.1)

Substituting in (3.1), the 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 SBTPPLD are obtained as

(3.2)

(3.3)

(3.4)

(3.5)

Using the relationship between moments about mean and the moments about origin, the moments about mean of SBTPPLD can be obtained as

(3.6)

(3.7)

(3.8)

The coefficient of variation, coefficient of skewness, coefficient of kurtosisand index of dispersion of SBTPPLD are thus given by

(3.9)

(3.10)

(3.11)

(3.12)

It can be seen that at these expressions reduce to the respective expressions of the SBPLD (1.1).

4. Estimation of Parameters

4.1. Maximum Likelihood Estimates

Let be a random sample of size n from the SBTPPLD (2.2) 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 SBTPPLD (2.2) is given by

(4.1)

and so the log likelihood function is obtained as

(4.2)

The two log likelihood equations are thus obtained as

(4.3)

(4.4)

The two equations (4.3) and (4.4) do not seem to be solved directly. However, the Fisher’s scoring method can be applied to solve these equations. We have

(4.5)

(4.6)

(4.7)

For the maximum likelihood estimates of of SBTPPLD (2.2), following equations can be solved

(4.8)

where and being the initial values of and are given by the method of moments. These equations are solved iteratively till sufficiently close estimates of and are obtained.

4.2. Estimates from Moments

The SBTPPLD has two parameters to be estimated and so the first two moments about origin are required to get the estimates of its parameters by the method of moments.

From (3.2) and (3.3) we have

(4.9)

Taking , we get

(4.10)

which gives

(4.11)

Replacing the first two population moments by the respective sample moments in (4.9) an estimate k of K can be obtained and using it in (4.11), an estimate of can be obtained.

Again, substituting in (3.2) and replacing the population mean by the sample mean and by , moment estimate of is obtained as

(4.12)

and so the moment estimate of is obtained as

(4.13)

5. Goodness of Fit

The SBTPPLD has been fitted to a number of data sets related to a number of observations of the size distribution of ‘freely –forming’ small groups in various public places reported by James (1953), Coleman and James (1961) and Simonoff (2003), and it was found that to almost all these data sets, the SBTPPLD provides closer fit than SBPD and SBPLD. Here, the goodness of fit of the SBTPPLD to three such data sets has been presented along with the goodness of fit given by SBPD and SBPLD.

The expected frequencies according to the SBPD and SBPLD have also been given in these tables for ready comparison with those obtained by the SBTPPLD. The estimates of the parameters have been obtained by the method of maximum likelihood estimation. On the basis of the values of chi-square, it can be seen that the SBTPPLD gives much closer fit than those by the SBPD and SBPLD.

The values of coefficient of variation (C.V), coefficient of skewness , coefficient of kurtosis and index of dispersion for estimated values of parameters for SBPD, SBPLD, and SBTPPLD and for original data for tables 1, 2 and 3 are presented in the following table 4.

Table 1. Counts of groups of people in public places on a spring afternoon in Portland

Table 2. Counts of Shopping Groups-Eugene, Spring, Department Store and Public Market

Table 3. Counts of Play Groups-Eugene, Spring, Public Playground

Table 4. Values of C.V,       of SBPD, SBPLD, and SBTPPLD for estimated values of parameters and for original data

It is also obvious from the analysis of the table 1, 2, 3, and 4 that the distribution which gives better fit in terms of chi-square values are the distribution whose index of dispersion for given values of the parameters is equal or nearer to the value of the index of dispersion of the original data. For example, it is clear that the index of dispersion of SBTPPLD is almost equal to the index of dispersion of the original data. Therefore, both the index of dispersion and the values of chi-square of SBTPPLD certify that the SBTPPLD is the best model than both SBPD and SBPLD for modeling data which structurally excludes zero counts.

6. Conclusions

In this paper, a size-biased two parameter Poisson-Lindley distribution (SBTPPLD), of which the size-biased Poisson-Lindley distribution (SBPLD) is a particular case, has been introduced to model count data which structurally excludes zero counts. The first four moments about origin, moments about mean, and expressions for coefficient of variation, skewness, kurtosis and index of dispersion have been obtained. The estimation of its parameters has been discussed using the method of maximum likelihood and the method of moments. The goodness of fit of the distribution has been presented to three data sets and it has been found that to all these data sets it provides much closer fit than both SBPD and SBPLD. The SBTPPLD has been found more general in nature and wider in scope than SBPD and SBPLD. Since SBTPPLD provides much closer fit to the observed data sets than those provided by the SBPD and SBPLD, SBTPPLD should be preferred over SBPD and SBPLD for modeling count data sets which structurally excludes zero counts.

ACKNOWLEDGEMENTS

The authors are grateful to the Editor-In-Chief of the journal and the anonymous reviewer for constructive and helpful comments which lead to the improvement in the quality of the paper.