Statistical applications of the Poisson-binomial and conditional Bernoulli distributions

The distribution of Z(1) + ... + Z(N) is called Poisson-Binomial if the Z(i) are independent Bernoulli random variables with not-all-equal probabilities of success. It is noted that such a distribution and its computation play an important role in a number of seemingly unrelated research areas such as survey sampling, case-control studies, and survival analysis. In this article, we provide a general theory about the Poisson-Binomial distribution concerning its computation and applications, and as by-products, Ne propose new weighted sampling schemes for finite population, a new method for hypothesis testing in logistic regression, and a new algorithm for finding the maximum conditional likelihood estimate (MCLE) in case-control studies. Two Of our weighted sampling schemes are direct generalizations of the ''sequential'' and ''reservoir'' methods of Fan, Muller and Rezucha (1962) for simple random sampling, which are of interest to computer scientists. Our new algorithm for finding the MCLE in case-control studies is an iterative weighted least squares method, which naturally bridges prospective and retrospectiue GLMs.