A GENERALIZATION OF SIZE-BIASED POISSON-SUJATHA DISTRIBUTION AND ITS APPLICATIONS

In this paper, a generalization of size-biased Poisson-Sujatha distribution (AGSBPSD) which includes both the size-biased Poisson-Lindley distribution (SBPLD) and the size-biased Poisson-Sujatha distribution (SBPSD) as particular cases has been proposed and studied. Its moments based measures including coefficients of variation, skewness, kurtosis, and index of dispersion have been derived and their shapes have been discussed with varying values of the parameters. The estimation of its parameters has been discussed using maximum likelihood estimation. Some applications of the proposed distribution have been explained using two count datasets.


INTRODUCTION
Size-biased distributions are a particular class of weighted distributions which arise naturally in practice when observations from a sample are recorded with probability proportional to some measure of unit size. In applications, size-biased distributions can arise either because individuals are sampled with unequal probability by design or because of unequal detection probability. Size-biased distributions come into play when organisms occur in groups, and group size influences the probability of detection. It was Fisher [1] who firstly introduced these distributions to model ascertainment biases which were later reformulated by Rao [2].
Size-biased distributions have applications in environmental science, econometrics, social science, biomedical science, human demography, ecology, geology, forestry etc. A summary of weighted distributions and size-biased distributions and their applications are available in Patil and Rao ([3], [4]).
Suppose a random variable X has probability distribution ( ) The modeling and statistical analysis of count data excluding zero counts using size-biasing of  [5] proposed negative binomial distribution (NBD) as a Poisson 6813 GENERALIZATION OF SIZE-BIASED POISSON-SUJATHA DISTRIBUTION mixture of gamma distribution and size-biased version of NBD were studied by Mir and Ahmad [6] whereas Sankaran [7] proposed the discrete Poisson-Lindley distribution (PLD) as a Poisson mixture of Lindley [8] distribution and size-biased version of PLD were studied by Ghitany and Al-Mutairi [9]. During recent decades several one parameter and two-parameter lifetime

Mixture of
Poisson and AGSD Shanker and Shukla [15] 6815 GENERALIZATION OF SIZE-BIASED POISSON-SUJATHA DISTRIBUTION

Shanker and
Shukla [17] The main motivation of this paper is to introduce a generalization of size-biased Poisson-Poisson

A GENERALIZATION OF SIZE-BIASED POISSON-SUJATHA DISTRIBUTION
Using (1.1) and the mean and the pmf of the AGPSD, the pmf of a generalization of size-biased Poisson-Sujatha distribution (AGSBPSD) can be defined as SHANKER, SHUKLA, TIWARI, LEONIDA It can be easily verified that AGSBPSD reduces to SBPSD and SBPLD at 1 respectively. Since some of the characteristics of AGSBPSD including coefficients of variation, skewness, kurtosis and index of dispersion depend on central moments, it has been observed that it difficult and very complicated to obtain the moments of AGSBPSD from its pmf directly.
Therefore, to obtain the moments of AGSBPSD easily, it has been observed that AGSBPSD can also be obtained as mixture of size-biased Poisson distribution (SBPD) with size-biased version of the generalization of Sujatha distribution (SBGSD).
Suppose X follows SBPD with parameter  having pmf and the parameter  follows SBGSD with parameters  and  having pdf

GOODNESS OF FIT
In this section the goodness of fit of SBPD, SBPLD SBPSD, GSBPLD and AGSBPSD has been presented for three count data-sets. The goodness of fit of these distributions are based on maximum likelihood estimates of the parameter. The first dataset in table 4 is the number of counts of pairs of running shoes owned by 60 members of an athletic club, reported by Simonoff [18] and the second dataset in table 5 is the distribution of free-forming small group size available in James [19] and Coleman and James [20]. It is clear from these two tables that AGSBPSD gives best fit in

ACKNOWLEDGMENT
Authors are thankful to the Chief Editor and referee for given valuable comments to the improve the quality of papers