On the use of asymptotic expansion in computing the null distribution of page's L-statistic

Abstract

Suppose that each out of n randomized complete blocks is obtained by observing a jointly continuous random variable taking values in R^k. Page's L-statistic is given then as a sum of i.i.d. lattice variables with finite moments of any order. Applying a well-known theorem on asymptotic expansions for the distribution function of such a sum yields higher order approximations to the significance probability of any observed value of L. The formula obtained by discarding terms smaller than o(n ^–1) is still very simple to use. Yet, due to it's strong analytical basis, it can be expected to provide substantial improvement on the traditional normal approximation. The results of extensive numerical investigations checking on both methods of approximation by comparison with the exact null distribution of L, prove this expectation fully warranted.

Keywords:

• lattice distribution
• exact p-values
• convolution algorithm
• approximation error
• Spearman's rank correlation coefficient