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

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.

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 |