Asymptotic Methods in Change-Point Analysis

This thesis is focussed on two areas of statistics, change-point analysis and functional data analysis, and the intersection of these two areas, i.e., the detection of structural breaks in functional stochastic models. The considered problems from (scalar) change-point analysis result from criticism of already existing sequential change-point procedures. The subject of this criticism was a lack of stability of these procedures regarding the time of occurrence of a change. As a possible solution to this criticism sequential methods are presented in this thesis in the framework of a linear regression model on the basis of the so-called Page CUSUM. To prove the desired properties of these procedures theoretically the asymptotic distribution of the delay time in the detection of structural breaks is derived in the special case of a location model. The notion "functional data analysis" represents an area of statistics that considers functions (in general defined on a compact interval) as "data points". Examples for such data are temperature curves or the path of a stock price on one trading day. To derive statistical procedures for this class of data dimension reduction techniques play a key role. The methods presented in this thesis are based on one of those techniques, the functional principal component analysis. They illustrate the construction of two-sample tests as well as change-point tests exploiting the properties of these functional principal components. In particular the problem of an adequate choice of the dimension of the space to project on in order to reduce the dimension is addressed and included in the construction of the respective testing procedures