Non-parametric, non-sequential change -point analysis

This dissertation is concerned with extending the non-parametric at most one change in the variance problem studied by E. Gombay, L. Horvath and M. Husukova [17] for the at most one change in variance case. The extension parallels the geometric method that was developed by M. R. Orasch in [23], [24] for constructing U-statistics based tests which can be used to detect multiple changes in the distribution of independent observations. The extension in Chapters 4 and 6 is achieved via results concerning weighted approximations of partial sum processes due to B. Szyszkowicz [31]--[34]. In particular a special case of B. Szyszkowicz [34, Theorem 2.1] is extended in the context of [17].;Chapter 1 provides many well-known results which are detailed in some definitions, lemmas and theorems. They figure prominently in the proofs of many of the theorems, lemmas and propositions to come.;Chapter 2 begins with a description of the generic change-point problem, and details some practical situations where change-point analysis can be used. This then leads to the statistical parameterization of change-point analysis and the construction of partial sum and U-statistic processes.;Chapter 3 introduces the at most two change-point partial sum process together with a detailed study of the asymptotic properties of certain functionals of this process under different assumptions. The asymptotic behaviour of these processes is studied under both the null H2O and the alternative H2A .;Chapter 4 repeats the results established in chapter 3 but for weighted partial sum processes. In this case, asymptotic results are established only under H2O and not H2A . It is expected that the test statistics constructed, as functionals of this process, should have greater power for detecting change-points on the tails than their unweighted counterparts.;Chapter 5 extends results developed in chapter 3 to the at most s change-points in variance setting. Again, the limiting behaviour of certain functionals of the developed partial sum process is studied under both HsO and HsA .;Chapter 6 extends the results developed in Chapter 4 regarding weights to the at most s changes in variance setting. Again, the behaviour is analyzed through a number of theorems but only under HsO .;Chapter 7 tabulates the limiting distribution of a number of change in mean statistics developed therein that are based on M. Csorgo&huml; and L. Horvath [5] via B. Szyszkowicz [34, Theorem 2.1], and can be used in testing the AMOC in mean hypothesis. Then a detailed Monte Carlo study is conducted where the power of many competing test statistics for the at most one change in mean hypothesis is explored. This chapter is founded on [25], a manuscript in preparation with Markus R. Orasch.