Note on Moments of a Logistic Order Statistics

Abstract

For a logistic distribution which is defined by $x = \ln \{F/(1 - F)\}$, where $F$ is the probability of a value less than $x$, Plackett [4], Birnbaum and Dudman [1], and Gupta and Shah [2], obtained the expressions for the moments of the ordered statistics in terms of digamma functions. Once the digamma functions are tabulated and their values are stored in the computer memory, it is easy to calculate the moments (first and second). This procedure was adopted by Birnbaum and Dudman and the author (jointly with others) in the past to calculate the moments of these order statistics for which some tables are available in the literature. In this note a recurrence relation for the moments of all order statistics for a logistic distribution is derived. This result not only generates higher moments of order statistics but also generates the moments of all order statistics for a logistic distribution, and this eliminates problems such as errors involved in computing and machine time etc. Thus using a computer language such as Fortran or APL, this simple relation is easily programmable for robustness studies, [3] for a logistic distribution.