Polya‐Eggenberger Distribution: Parameter Estimation and Hypothesis Tests

Abstract

SummaryAlmost all common discrete distributions are related to the Polya-Eggenberger distribution (PED), either ita special cases or ita limiting distributions. We demonstrate that the sum of n binary random variables Yj (j= 1, ..., n) taking values of 0 or 1 follows a P E D if and only if the conditional expectation of Ya with rsapect to Y1, ..., Ya-1 is a linear fanotion of Yi, ..., Yr-1, the expectatione E Yj (j= 1, ..., n) are the same, and for each pair T i and Yj, the correlations are the same.The maximum likelihood estimation of the parameters is studied. In most cases, the maximum likelihood equations can be solved by the Newton-Rapheon iterative procedure; in a special case. the maximum likelihood parameter mtimates can be expreased as a function of the observed frequencies; and in some cnses, the maximum likelihood equatione are not soluble. Even when the maximum likelihood equations are soluble, the solutions may not be permissible. We propose a method to handle this problem.For testing the hypothesis that the parameter is zero, the Wald statistic is used; for model selection, the likelihood ratio test is used. The hypothesis testa are described by two data examples and applications of the P E D to data analysis are demonstrated.

Run and Frequency Quotas Under Markovian Fashion and their Application in Risk Analysis

“…Qu et al (1990), there exists its discrete analogue in terms of "distribution of order k". The distributions of order k retain the mutual relationships among the class of Polya-Eggenberger set of distributions.…”