SADDLEPOINT APPROXIMATIONS TO THETWO-SAMPLE PERMUTATION TESTS

1.IntroductionPermutationtestisapowerfultoolinstatisticalinference.However,itsapplicationishinderedbythecomputationalintensity.Onewaytoovercomethisdifficultyistouseasymptoticapproximations.EdgeworthexpansionshavebeenobtainedbyAlbers,BickelandVanZ..tti]jBickelandVanZwetZItRobinsonlg]tJohnandR.bi....lo]tandthesegivegoodapproximationsinmostcases.However,theseEdgeworthexpansionssometimescangiveveryinaccurateresultsinthetailsofthedistribution,withwhichwearemostconcernedforinferenceandforcon…

Saddlepoint approximations to the two-sample permutation tests

A saddlepoint approximation for a two-sample permutation test was obtained by Robinson7].Although the approximation is very accurate, the formula is very complicated and difficult toapply. In this papert we shall revisit the same problem from a different angle. We shall first turnthe problem into a conditional probability and then apply a Lugannani-Rice type formula to it,which was developed by Skovagard8] for the mean of i.i.d. samples and by Jing and Robinson5]for smooth function of vector means. Both the Lugannani-Rice type formula and Robinson'sformula achieve the same relative error of order O(n-3/2), but the former is very compact andmuch easier to use in practice. Some numerical results will be presented to compare the twoformulas.