Nonparametric estimation of change-points in derivatives

Abstract

In this thesis, the main concern is to analyse change-points in a non-parametric regression model. More specifically, the analysis is focussed on the estimation of the location of jumps in the first derivative of the regression function. These change-points will be referred to as kinks. The estimation method is closely based on the zero-crossing technique (ZCT) introduced by Goldenshluger, Tsybakov and Zeevi (2006). The work of Goldenshluger et al. (2006) was aimed at estimating jumps in the regression function in the indirect non-parametric regression model and shown to be optimal in the minimax sense. Their analysis was applied in practice by Cheng and Raimondo (2008) whereby a class of kernel functions is constructed to use ZCT with a kernel smoothing implementation. Moreover, Cheng and Raimondo (2008) adapted the technique to estimating kinks from a fixed design model with i.i.d. errors. The thesis extends the aforementioned kink estimation technique in two ways. The first extension is to include a long-range dependent (LRD) error structure in the fixed design scenario. The rate of convergence of the resultant LRD method is shown to be reliant on the level of dependence and the smoothness of the underlying regression function. This rate of convergence is shown to be optimal in the sense of the minimax rate. The second extension is to include a regression model with random design and LRD structures. The random design regression models considered include an i.i.d. random design with LRD errors and a separate model with a LRD design with i.i.d. errors. For the case of LRD design variables, the rate of convergence for the estimator is again reliant on the level of dependence and the smoothness of the regression function. However, interestingly for the case of i.i.d. design and LRD errors, the rate of convergence is shown to not rely on the level of dependence but only rely on the smoothness of the regression function and in fact agrees with the minimax rate for fixed design with i.i.d. errors. To conclude, it is summarised where original work occurs in this thesis. Firstly, the extension of the ZCT to the fixed design framework with LRD noise arose with discussions with my initial Ph.D. supervisor Dr Marc Raimondo before his passing. The method is based on the technique proposed by Cheng and Raimondo (2008) but the mathematical analysis and development of the extension to the LRD framework and its minimax optimality is my own work. For the second extension which covers the random design regression framework, the main idea and premise arose through discussions with Assistant Professor Rafal Kulik. I wish it to be known that although the published versions of the work are in joint names with Assistant Professor Kulik, the great bulk of the mathematical analysis and development presented in this thesis is my own. Finally my current supervisor's contribution, Professor N. C. Weber, has been to provide direction in terms of checking the accuracy, clarity and style of the work.