Contributions to the m–out–of–n bootstrap

Abstract

The traditional bootstrap is a sophisticated resampling procedure that has received a great deal of attention in the literature over the past three decades. This thesis focuses on a variation of the traditional bootstrap, called the m-out-of-n bootstrap, where one resamples fewer than n observations from a sample of size n. It is shown, by referring to various sources in the literature, that this modification of the traditional bootstrap has many desirable properties, not least of which is that it rectifies certain inconsistencies suffered by the traditional bootstrap. The aim of thesis is twofold: First, to explore the collection applications of the m-out-of-n bootstrap and examine their usefulness, and second, to contribute to collection by developing new applications, providing the necessary tools to apply them correctly, and to obtain estimators for the resample size, m. The few chapters of this thesis are a literature study which examines the development of the theory underlying the traditional bootstrap and the m-out-of-n bootstrap as well as considering the practical applications of these two techniques. Included is a discussion of situations where the traditional bootstrap method fails to produce consistent results, but where the m-out-of-n bootstrap is consistent under minimal conditions. Once the basic theoretical background of the m-out-of-n bootstrap has been established, a new methodology for applying the m-out-of-n bootstrap in point estimation problems is presented. A contrast is made between the naive application of the m-out-of-n bootstrap and the new methodology by referring to the new method as the ‘corrected m-out-of-n ‘ bootstrap. The use of the m-out-of-n bootstrap is considered two new areas of application: • First, a new method for point estimation of parameters based on BRAGGing (Bootstrap Robust AGGregating) estimation methods is proposed both the original naive m-out-of-n bootstrap methodology, as well as the newer, corrected m-out-of-n bootstrap methodology. The estimation of the resample size for this estimation problem is also addressed by considering Cornish-Fisher and other expansions. • Second, the application of the m-out-of-n bootstrap to hypothesis testing is considered. Two new data-based choices of the resample size, m, are proposed in this setup. The first estimator is based on a bootstrap estimate of the size of the test using a bootstrap critical value, and the second is based on the probability structure of the p-values of a test under the null hypothesis.

In both of these new areas of application, the data-dependent choices of m are theoretically and numerically motivated, the former being accomplished through the use of comprehensive mathematical arguments and the latter through the use of extensive Monte-Carlo simulations.