Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements

Abstract Approximate tests for a composite null hypothesis about a parameter θ may be obtained by referring a test statistic to an estimated critical value. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. The simple asymptotic test ϕA refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated. The bootstrap approach used here is based on the concept of prepivoting. Prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution. The simple bootstrap test ϕB refers the prepivoted test statistic to a quantile of the uniform (0, 1) distribution. Under regularity conditions, the bootstrap test ϕB has a smaller asymptotic order of error in level than does the asymptotic test ϕA , provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters. In the contrary case, both ϕA and ϕB have the same order of level error. Certain classical refinements to asymptotic tests can be regarded as analytical approximations to the bootstrap test ϕB. These classical results include Welch's estimated t distribution solution to the Behrens-Fisher problem, Bartlett's adjustment to likelihood ratio tests, and Edgeworth expansion corrections to the nonparametric t test for a mean. On the other hand, the bootstrap test ϕB can also be approximated directly by a Monte Carlo algorithm. The prepivoted bootstrap test ϕ B,1 is obtained by prepivoting the test statistic twice before referring the result to a quantile of the uniform (0, 1) distribution. Under regularity conditions, the level error of ϕ B,1 is of smaller asymptotic order than is the level error of either ϕB or ϕA. Analytical and semianalytical approximations sometimes exist for ϕ B,1. In general, a nested double-bootstrap Monte Carlo algorithm yields a satisfactory approximation to ϕ B,1. The possibility of direct nonanalytical implementation is a great practical merit of both ϕB and ϕ B,1.