Saddlepoint approximations for the difference of order statistics

Abstract

SUMMARY Saddlepoint approximations for the distribution of the difference of order statistics from a continuous distribution are presented. The approximations are illustrated for the case of the exponential distribution where we can find the exact probability density function of the difference of order statistics, and for the case of the noncentral chi-squared distribution which has application to the problem of testing a time series for linearity. In both cases we show that the saddlepoint approximation is much better than the normal approximation, especially for realistic sample sizes.