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 class...

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