Moment-Based Approximations of Probability Mass Functions with Applications Involving Order Statistics

Abstract

It is shown in this article that a technique that was previously introduced to approximate the density functions of certain continuous random variables can be successfully applied to discrete distributions. The probability mass function approximants are expressed as the product of an appropriate base density function and a polynomial adjustment. A degree selection criterion that is based on the integrated squared difference between approximants of successive degrees is being proposed. The methodology, which is conceptually simple and easily implementable, is applied to a binomial random variable, the largest order statistic in a binomial sample, a Poisson distribution, and two rank-sum test statistics.