Some aspects of saddlepoint approximations for non-parametric statistics
Rungao Jin · Bulletin of the Australian Mathematical Society · 2001
Some aspects of saddlepoint approximations for non-parametric statistics RUNGAO JIN This thesis is concerned with Edgeworth and saddlepoint approximations and robust permutation tests in which the saddlepoint approximation has been used.There are five chapters in this thesis.The first chapter is the introduction.The next three chapters are concerned with Edgeworth and saddlepoint approximations.The last chapter is about robust permutation tests in which the large deviation results related to the saddlepoint approximation have been used.Chapter 2 contains saddlepoint approximations to the probability and probability density and tail probability of X, the mean of a sample from a finite population.A unification of work based on the conditional saddlepoint and Edgeworth approximations is obtained although there are some original contributions in the details of the proof.In Section 1 of Chapter 2, the lattice situation is considered.In Section 2, the general case is considered.In Chapter 3 the saddlepoint approximations near the endpoints of the support are obtained for binomial, hypergeometric, one and two sample Wilcoxon statistics.The usual saddlepoint approximations of Daniels (1954) are based on asymptotic normality of tilted means.However, if the means have finite support then the approximations can fail near the endpoints of the support where asymptotic normality of the tilted means does not hold.This chapter gives new accurate methods for saddlepoint approximations at the extremes of the support and new relative error rates are obtained near the endpoints for these statistics.The new approximations are compared numerically to those based on the normal saddlepoint method for one-sample and two-sample Wilcoxon statistcs.In Chapter 4 a new saddlepoint approximation is obtained based on the exact moment generating function of van Dantzig (1947van Dantzig ( -1950)).These approximations are compared numerically to those based on the conditional saddlepoint method and to the method of Froda and van Eeden [1] for n = m = 5 and m = 10, n = 6.In general, we see the new method is a little better than the conditional method but not at all points,