Mathematics Ph.D. Dissertations

Energy-Statistics-Based Nonparametric Tests for Change Point Analysis

Date of Award

2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Wei Ning (Committee Chair)

Second Advisor

John Chen (Committee Member)

Third Advisor

Junfeng Shang (Committee Member)

Abstract

In our research, we exploit the relationship between properties of U-statistics and Energy statistics (V-statistics) to come up with non-parametric tests in change-point analysis. \cite{lee} provided a wide discussion on asymptotic behaviour, and connection between U-statistics and V-statistics when large samples. Many other researchers such as \cite{sen1974}, \cite{serfling1980} and \cite{neuhaus1977} studied connections between U-statistic and V-statistics and their asymptotic properties. We first propose a non-parametric test to detect change in the distribution based on MIC using energy statistics. The proposed energy-statistic based MIC is used for model selection between null and alternative hypothesis models. We achieve this by adopting the idea of the works of \cite{chen} and \cite{pan} and apply energy distance statistic. To test the performance of our proposed test, we assess the finite sample properties and compare efficiency and powers of different methods with that of our method. We then discuss applications of our proposed test in two different real-life examples for detecting change in mean and variance, respectively. Since the underlying distribution is unknown, we use bootstrap approximations for the p-values as proposed by \cite{hangfen2009} in detecting unknown change points in means and variances. In the second part of my dissertation, we propose a non-parametric sequential test based on energy statistics \cite{rizzo2013} to detect changes in distribution for independent random variables. In their study, \cite{Oscar} considered backward-looking windows each of length $L$ across the pooled data, and then retrospectively investigate if there is evidence for a change point between the times $\text{max}\{t-L,1\}$ and $t$, for any given time $t$. We adopt this idea to come up with a test statistic similar in structure based on energy statistics. We compare the performance of this method in terms of false-alarm rates and powers to existing sequential methods such as sequential KS, Generalized Likelihood Ratio test and others for detecting change in distribution. We apply our proposed method and others to the problem of detecting radiological anomalies.

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