Adaptive Parametric Change Point Inference Under Covariance Structure Changes

Vasileios Pavlopoulos

doi:10.7273/000005314

The thesis provides an in-depth and comprehensive exploration of change point analysis, delving into various critical procedures and focusing on change point estimation. The first part of the thesis serves as a robust foundation for the subsequent research, consisting of an extensive literature review on change point analysis. While the primary objective is to provide a comprehensive overview of the existing literature on change point estimation and detection, it goes beyond a mere survey by presenting novel methodologies that contribute to the general framework of change point analysis in various directions. By thoroughly examining the existing body of knowledge, I establish a comprehensive understanding of the field, enabling the research to build upon and contribute to the existing literature. This meticulous examination of prior works ensures that the subsequent methodologies and findings are firmly grounded in change point analysis's current state of the art.In the second part, the focus shifts towards developing three novel methodologies for change point estimation. These methodologies aim to surpass the limitations of existing models, particularly in the context of covariance matrix for financial time series data. By employing a change point approach, the proposed methods outperform traditional varying-coefficient parametric models like GARCH, as the coefficients in these models may vary with time. In chapter 2, sufficient conditions are obtained under which the maximum likelihood process is adaptive against the covariance estimate to yield an optimal rate of convergence with respect to the change size. This rate is preserved while allowing the jump size to diminish. Under these circumstances, argmax results of a two-sided negative Brownian motion or a two-sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively, provide inference for the change point parameter. Theoretical results are supported by the Monte-Carlo simulation study. A bivariate data on daily log returns of two stock market indices as well as tri-variate data on daily log returns of three banks are analyzed by constructing confidence interval estimates for multiple change points that have been identified previously for each of the two data sets, providing insights into the underlying dynamics. In the third part of the thesis, the application of change point estimation in evaluating the performance of existing detection algorithms is explored. By incorporating the estimated change points, confidence interval estimates are constructed for multiple identified change points within a twelve-dimensional dataset. This analysis contributes to the understanding of how change point estimation can enhance the detection algorithms' effectiveness. The proposed methods offer improved accuracy and performance, supported by theoretical analysis and empirical studies using real-world datasets.

Adaptive Parametric Change Point Inference Under Covariance Structure Changes

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