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 ϕ_A 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 ϕ_B refers the prepivoted test statistic to a quantile of the uniform (0, 1) distribution. Under regularity conditions, the bootstrap test ϕ_B has a smaller asymptotic order of error in level than does the asymptotic test ϕ_A , provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters. In the contrary case, both ϕ_A and ϕ_B have the same order of level error. Certain classical refinements to asymptotic tests can be regarded as analytical approximations to the bootstrap test ϕ_B. 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 ϕ_B 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 ϕ_B or ϕ_A. 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 ϕ_B and ϕ _B,1.

Key Words:

• Asymptotic test
• Bartlett adjustment
• Behrens-Fisher problem
• Bootstrap test
• Edgeworth expansion
• Error in rejection probability