Resampling Methods for the Change Analysis of Dependent Data

Abstract

The fundamental question in change-point analysis is whether an observed stochastic process follows one model or whether the underlying model changes at least once during the observational period. Most of the older works discuss independent observations, yet from a practical point of view cases of dependent data have become more and more important. In this dissertation we develop testing procedures for dependent models. In change-point analysis critical values for testing procedures are usually obtained by distributional asymptotics. These critical values, however, do not sufficiently reflect dependency. Moreover it is a well-known fact that convergence rates especially for extreme-value statistics are very slow. Using resampling methods we obtain better approximations, which take possible dependency structures more efficiently into account. We prove that the original statistics and their resampling counterparts follow the same distributional asymptotics. First we obtain limit theorems for the corresponding rank statistics, which then combined with laws of large numbers imply the resampling asymptotics conditionally on the given data. In a first part we consider abrupt and gradual changes in models of possibly dependent observations satisfying a strong invariance principle. The main part of this dissertation studies a location model with dependent errors that form a linear process. Different types of statistics are considered, such as maximum-type statistics (particularly different CUSUM procedures) or sum-type statistics. The resampling-methods have to be adapted to allow for dependent errors. Thus, we analyze a block bootstrap as well as a bootstrap in the frequency domain. Finally, some simulation studies illustrate that the permutation tests usually behave better than the original tests if performance is measured by the type I and II errors, respectively.

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