Bootstrap-based hypothesis testing

Thesis (Ph.D. (Statistics))--North-West University, Potchefstroom Campus, 2009.

One of the main objectives of this dissertation is the development of a new method of evaluating the performance of bootstrap-based tests. The evaluation method that is currently in use in the literature has some major shortcomings, for example, it does not allow one to determine the robustness of a bootstrap estimator of a critical value. This is because the evaluation and the estimation are based on the same data. This traditional method of evaluation often leads to too optimistic probability of type I errors when bootstrap critical values are used. We show how this new, more robust, method can detect defects of bootstrap estimated critical values which cannot be observed if one uses the current evaluation method. Based on the new evaluation method, some theoretical properties regarding the bootstrap critical value are derived when testing for the mean in a univariate population. These theoretical findings again highlight the importance of the two guidelines proposed by Hall and Wilson (1991) for bootstrap-based testing, namely that re-sampling must be done in a way that reflects the null hypothesis and bootstrap tests should be based on test statistics that are pivotal (or asymptotically pivotal). We also developed a new non-parametric bootstrap test for Spearman's rho and, based on the results obtained from a Monte-Carlo study, we recommend that this new test should be used when testing for Spearman's rho. A semi-parametric test based on copulas was also developed as a useful benchmark tool for measuring the performance of the non-parametric test. Other research objectives of this dissertation include, among others, a brief overview of the non-parametric bootstrap and a general formulation of methods which can be used to apply the bootstrap correctly when conducting hypothesis testing.

Doctoral