Polya‐Eggenberger Distribution: Parameter Estimation and Hypothesis Tests

Abstract

AbstractAlmost all common discrete distributions are related to the Polya‐Eggenberger distribution (PED), either its special cases or its limiting distributions. We demonstrate that the sum of n binary random variables Yj (j=1, …, n) taking values of 0 or 1 follows a PED if and only if the conditional expectation of Yk with respect to Y1, …, Yk is a linear function of Y1, …, Yk‐1, the expectations E Yj (j = 1,…, n) are the same, and for each pair Yi 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‐Raphson iterative procedure; in a special case, the maximum likelihood parameter estimates can be expressed as a function of the observed frequencies; and in some cases, the maximum likelihood equations 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 tests are described by two data examples and applications of the PED to data analysis are demonstrated.