Mathematics Ph.D. Dissertations

Title

Sequential Change-point Detection in Linear Regression and Linear Quantile Regression Models Under High Dimensionality

Date of Award

2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Wei Ning (Advisor)

Second Advisor

Andy Garcia (Other)

Third Advisor

Hanfeng Chan (Committee Member)

Fourth Advisor

Junfeng Shang (Committee Member)

Abstract

Sequential change point analysis aims to detect structural change as quickly as possible when the process state changes. A good sequential change point detection procedure is expected to minimize the detection delay time and the risk of raising false alarm. Existing sequential change point detection methods cannot be applicable for high-dimensional data because they are univariate in nature and thus present challenges.

In the first part of the dissertation, we develop a monitoring method to detect structural change in smoothly clipped absolute deviation (SCAD) penalized regression model for high-dimensional data after the historical sample with the sample size m. The unknown pre-change regression coefficients are replaced by the SCAD penalized estimator. The asymptotic properties of the proposed test statistics are derived. We conduct a simulation study to evaluate the performance of the propose method. The proposed method is applied to the gene expression in the mammalian eye data to detect changes sequentially.

In the second part of the dissertation, we develop a sequential change point detection method to monitor structural changes in SACD penalized quantile regression (SPQR) model for high-dimensional data. We derive the asymptotic distributions of the test statistic under the null and alternative hypotheses. Furthermore, to improve the performance of the SPQR method, we propose the Post-SCAD penalized quantile regression estimator (P-SPQR) for high-dimensional data. Simulations are conducted under different scenarios to study the finite sample properties of the SPQR and P-SPQR methods. A real data application is provided to demonstrate the effectiveness of the method.

In the third and fourth part of the dissertation, we investigate the change point problem for Skew-Normal distribution and three parameter Weibull distribution respectively. Besides detecting and obtaining the point estimate of a change location, we propose an estimation procedure based on the confidence distribution (CD) along with the modified information criterion (MIC) to construct the confidence set for the change location. Simulations are conducted to evaluate the performance of the proposed method in terms of powers, coverage probabilities and average lengths of confidence sets. Real data applications are provided in each part to illustrate the performance of the proposed methods.

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