Mathematics Ph.D. Dissertations

Sequential Change-point Analysis for Skew Normal Distributions and Nonparametric CUSUM and Shiryaev-Roberts Procedures Based on Modified Empirical Likelihood

Date of Award

2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Wei Ning (Committee Chair)

Second Advisor

Erin Labbie (Other)

Third Advisor

John Chen (Committee Member)

Fourth Advisor

Junfeng Shang (Committee Member)

Abstract

Sequential change-point analysis identifies a change of probability distribution in an infinite sequence of observations generated by a process, by repetitively performing a hypothesis test each time a new observation is generated and added to the current data set. It has important applications in many fields, such as financial investment, system monitoring, and quality control. While lots of research have been done for different scenarios, especially time series, few works have been developed for skew data, as well as for the case where the distribution family of observations is unspecified. Hence, in this dissertation, we focus on developing a sequential point detection procedure for the skew-normal distribution family, and a nonparametric procedure based on the modification of previous methods. In the first part of the dissertation, we propose a sequential change-point detection rule for skew-normal distribution, by modifying the procedures proposed by Mei (2006). We focus on the change of location and shape parameters, respectively under the simple and composite alternative hypothesis. We derive the optimality of our modified procedure for location parameter under a simple alternative hypothesis. Also, the simulation shows that our new procedure has fewer false alarms than previous methods when the exact value of pre-change and post-change parameters are not specified. In the second part, we proposed a nonparametric sequential change-point detection procedure, by modifying Page’s CUSUM procedure and the well-known Shiryaev-Roberts (SR) procedure. More specifically, we substitute the parametric likelihood function in the two methods with empirical likelihood (EL), which allows us to perform a likelihood ratio test without knowing the distribution family. Also, we assume training data are available to estimate pre-change and post-change parameters. Different versions of empirical likelihood are applied and simulations are conducted to show their performance. The result shows that compared to previous approaches, such as kernel density estimation, our new method performs better and gives a smaller detection delay when the training sample is small and the underlying distribution is not specified. This is true for both EL-based CUSUM and EL-based SR. Also, EL-based methods are least impacted by the shrinkage of training samples.

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