Krawtchouk’s Polynomial for Hypergeometric Distribution Approximation

Abstract

A formula for the expansion of the hypergeometric probability approximation in terms of Krawtchouk’s polynomial was proposed—this formula is called a modified binomial probability—and its accuracy was investigated in terms of the total variation distance. In addition, an efficiency comparison with a binomial probability and Ord’s probability was conducted using a simulation study for 288 situations. It was found that the total variation distance of a modified binomial probability was less than those of the binomial probability and Ord’s probability for all situations and tended to zero for a small sampling fraction. Even for a large population size of 20,000, there seemed to be no difference in the efficiency of the modified binomial probability, Ord’s probability and the binomial probability when the sampling fraction was greater than 0.1 at all levels of the proportion of the population that had the specified attribute.