Nonparametric testing in modern statistics: A personal journey

Nonparametric testing is a fundamental branch of statistics and has numerous applications in modern statistics. This thesis is based on four projects in which we study four different nonparametric testing problems. In the first project, we study the two-sample test and aim to test the equality of two high-dimensional distributions. In particular, we propose a novel studentized test statistic based on the maximum mean discrepancy and establish the asymptotic theory of the proposed test in the high dimensional setting. In the second project, we investigate the change point testing problem for vector-valued time series with both temporal and cross-sectional dependence. By integrating the idea of sample splitting and self-normalization, we propose a dimension-agnostic testing method applicable to low-, medium-, and high dimensional settings, and provide the asymptotic properties of the proposed test both under the null and against the local alternatives. In the third project, we focus on functional time series inference and generalize our test proposed in the previous project to the infinite-dimensional setting. In particular, we propose a fully functional approach based on sample splitting and illustrate it for several testing problems. In the fourth project, we consider the two-sample conditional distribution test and propose a new population-level metric that characterizes the discrepancy between two conditional distributions. We also construct a test statistic using generative adversarial networks and introduce a multiplier bootstrap procedure to approximate the critical value. The asymptotic theory is provided under the null and against the local alternatives, and some preliminary simulations are presented to demonstrate the effectiveness of the test.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Statistics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Gao, Hanjia

Contributors dc:contributor

• Shao, Xiaofeng
• Yang, Yun
• Simpson, Douglas
• Wang, Yuexi

• Nonparametric Testing
• High-dimensional Statistics
• Functional Data
• Time Series

Statement dc:rights

• Copyright 2024 Hanjia Gao

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/125689

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F125689