Theses and Dissertations

Advisor

Wu, Tung-Lung

Committee Member

Popescu, George

Committee Member

Patil, Prakash

Committee Member

Woody, Jon

Committee Member

Zhang, Jialin; Chen, Xinyuan

Date of Degree

8-12-2026

Original embargo terms

Immediate Worldwide Access

Document Type

Dissertation - Open Access

Major

Mathematical Sciences

Degree Name

Doctor of Philosophy (Ph.D.)

College

College of Arts and Sciences

Department

Department of Mathematics and Statistics

Abstract

This dissertation studies scan statistics for detecting localized changes in non-homogeneous Poisson processes. The motivating problem arises from genomic sequencing data, where tumor and normal read locations may differ over short regions due to copy number variation, while the underlying event intensity varies across the domain. In this setting, methods based on homogeneous Poisson process assumptions are not appropriate. The dissertation develops and studies scan-based methods for detecting local differences while accounting for non-homogeneous structure. Three related approaches are considered. The first develops a scan statistic for a nonhomogeneous Poisson process and uses extreme value calibration after transformation under the null model, while accounting for the dependence structure of the scan process. The second considers the two-sample problem and constructs a permutation-based scan procedure after an empirical distribution transformation, yielding a conditional framework for testing whether the two samples arise from the same underlying distribution. The third develops a bias-corrected kernel transformation for the two-sample setting and applies the scan statistic to the transformed sample in order to reduce transformation error. The proposed methods are studied through simulation under linear and exponential intensity models, with performance assessed through Type I error and power across different window widths and signal strengths. The methods are also applied to tumor and normal sequencing data to assess their ability to identify localized changes consistent with copy-number variation. Overall, the dissertation shows how extreme-value calibration, permutation testing, and bias-corrected transformation can be used to adapt scan statistics to nonhomogeneous Poisson process data.

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