Polya-Eggenberger Distribution: Parameter Estimation and Hypothesis Tests

✍ Scribed by Yinsheng Qu; G. J. Beck; G. W. Williams

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 558 KB
• Volume: 32
• Category: Article
• ISSN: 0323-3847
• DOI: 10.1002/bimj.4710320215

✦ Synopsis

Almost 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.