Thesis Defence: Ranking Functions of Data

Cassie Yue Zhang, supervised by Dr. Xiaoping Shi, will defend their thesis titled “Ranking Functions of Data” in partial fulfillment of the requirements for the degree of Master of Science in Mathematics.

An abstract for Cassie Yue Zhang thesis is included below.

Defences are open to all members of the campus community as well as the general public. Registration is not required for in person defences.

---

ABSTRACT

This thesis addresses a significant problem in numerous scientific fields – the challenge of determining whether two sets of data are statistically identical, a procedure known as a two-sample homogeneity test.

Various methods, including well-known parametric tests like t-tests and ANOVA, have been employed for two-sample homogeneity tests. However, these tests have certain limitations. They rely heavily on specific assumptions about the data’s distribution. If these assumptions are violated, the results can be misleading.

To overcome these limitations, non-parametric methods that do not re- quire strict assumptions about the data’s nature have been explored. How- ever, these too have their shortcomings. For instance, techniques like the Wilcoxon rank-sum test can only be applied effectively in univariate cases, as multivariate data lacks a natural ranking. This complicates the analysis and limits the utility of such methods for multivariate data.

In response to these challenges, this thesis introduces the innovative concept of data depth, offering a new way to rank multivariate data. After a thorough comparison of eleven widely-used depth functions and the introduction of two new two-sample homogeneity tests. This advancement significantly improves the ability to conduct accurate two-sample homogeneity tests on multivariate data.

Further enhancing the power and potential of depth-based tests, two new test statistics are proposed for assessing the homogeneity of two multivariate samples. Inspired by the quality index used in quality assurance contexts, the construction of these statistics is based on pairwise quality indexes, leading to a χ²(1) asymptotic null distribution. Simulation studies further underline the promising performance of these proposed tests, with additional generalization into multivariate multisample situations also discussed.

The increasing number of samples correspondingly increases the number of pairwise quality indexes, an aspect considered and explored as part of the study. Future works include application on high-dimensional data and change point analysis. With its focus on boosting the power of depth-based tests, this thesis marks a significant step forward in the field of data analysis.