Bootstrap methods are based on generating virtual samples from a given data set. Therefore an underlying sample itself can influence the outcome of bootstrap procedures seriously. In the case of test procedures the bootstrap can be used to estimate critical values and power functions (Beran (1986)). In Wolf-Ostermann (1996) we have shown that the breakdown of the bootstrap-t-test can be characterized by empirical measures of the underlying sample. Based on these investigations we developed different (rough-) adaptive bootstrap tests for a location parameter which depend on the empirical skewness and kurtosis of the underlying sample (Wolf-Ostermann (1997)).
The basic idea of both bootstrap and adaptive methods is a strongly data dependent approach. Therefore it is suggestive and obvious to combine these methods in order to develop a (fine-) adaptive bootstrap test procedure which is strictly based on the underlying sample. We propose an approach for a location test in the one-sample case which uses the estimated power functions itself as a selection criterion to adapt the appropriate test. The approach is based on the following tests: the t-test, Johnson's modified t-test, a robust test based on M-estimates and two adaptive trimmed tests. The combination of bootstrap and adaptive methods for constructing test procedures in this way is a new approach which is not wide-spread in current statistical literature.
Since theoretical results for this approach can only be archieved asymptotically, we show that all involved bootstrap tests have asymptotic level alpha and the estimated power functions are uniformly or pointwise consistent. Our approach then defines a test which also has asymptotic level alpha and maximizes the estimated power function. Following Young (1994) who demands empirical investigations of bootstrap properties in the finite sample case, the performance of the proposed adaptive bootstrap test procedure is analysed by a simulation study. All comparisons are carried out by considering significance levels and powers. The research concentrates on the case of small and moderate sample sizes which are of interest for the practical use of bootstrap tests.