Small sample approximations for spacing statistics

Abstract

Edgeworth expansions as well as saddle-point methods are used to approximate the distributions of some spacing statistics for small to moderate sample sizes. By comparing with the exact values when available, it is shown that a particular form of Edgeworth expansion produces extremely good results even for fairly small sample sizes. However, this expansion suffers from negative tail probabilities and an accurate approximation without this disadvantage, is shown to be the one based on saddle-point method. Finally, quantiles of some spacing statistics whose exact distributions are not known, are tabulated, making them available in a variety of testing contexts. (~)

The distribution of the square sum of Dirichlet random variables and a table with quantiles of Greenwood's statistic

“…The computation of the distribution of U 2 n has a long history beginning with Moran (1947) and(1951), Gardner (1952) followed by Hill (1979), Burrows (1979), Currie (1981), Stephens (1981) Currie and Stephens (1986), Does et al (1988), Ghosh and Jammalamadaka (1998) and (2000). The exact methods of Burrows, Currie and Stephens use iterated integrations up to n = 20.…”