Prepivoting test statistics: A bootstrap view of asymptotic refinements. (English) Zbl 0662.62024

Beran, Rudolf

J. Am. Stat. Assoc. 83, No. 403, 687-697 (1988).

Approximate tests for a composite null hypothesis about a parameter \(\theta\) 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 \(\phi_ A\) 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 \(\phi_ B\) refers the prepivoted test statistic to a quantile of the uniform (0,1) distribution. Under regularity conditions, the bootstrap test \(\phi_ B\) has a smaller asymptotic order of error in level than does the asymptotic test \(\phi_ A\), provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters. In the contrary case, both \(\phi_ A\) and \(\phi_ B\) have the same order of level error.
Certain classical refinements to asymptotic tests can be regarded as analytical approximations to the bootstrap test \(\phi_ B\). 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 \(\phi_ B\) can also be approximated directly by a Monte Carlo algorithm.

MSC:

62F05	Asymptotic properties of parametric tests
62E20	Asymptotic distribution theory in statistics
62F03	Parametric hypothesis testing

Keywords:

error in rejection probability; Approximate tests; composite null hypothesis; estimated critical value; bootstrap methods; prepivoting; bootstrap null distribution; asymptotic null distribution; asymptotic tests; analytical approximations; Welch’s estimated t distribution solution to the Behrens-Fisher problem; Bartlett’s adjustment to likelihood ratio tests; Edgeworth expansion corrections