Moment relations for order statistics of the standardized gamma distribution and the inverse multinomial distribution

Abstract

In inverse multinomial sampling, observations are selected one at a time from a multinomial distribution with k cells and associated probabilities il1, . . ., 11k, where Z HII = 1. The order statistics represent the numbers of multinomial observations required to obtain a predetermined number of observations, A, say, from any one of the k cells, from each of any two of the cells, and so on. The smallest order statistic is of particular interest since it is used as a sampling termination criterion in a selection procedure proposed by Cacoullos & Sobel (1966), for determining the most probable category in a multinomial distribution. The maximum expected amount of sampling occurs for the configuration Hi = k-1 (i 1 l,k), which we refer to as the symmetrical case. In this note, we establish a very simple relation between moments of the distributions of the order statistics of the symmetrical inverse multinomial distribution and the order statistics of independent standardized gamma variables with integer parameter A. Gupta (1960) considered the order statistics of the gamma distribution with integer parameter and presented tables of the first four moments of their distributions for A = 1 (1) 5 and k = 1 (1) 10. We also present and compare some asymptotic expressions for the moments.