Mathematics Ph.D. Dissertations

Projection Pursuit Indices Based on Weighted L2 Statistics for Testing Normality

Date of Award

2023

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics/Mathematical Statistics

First Advisor

Maria Rizzo (Committee Chair)

Second Advisor

Angela Nelson (Other)

Third Advisor

Umar Islambekov (Committee Member)

Fourth Advisor

Wei Ning (Committee Member)

Abstract

Projection pursuit is the process of finding interesting d-dimensional projections from an n × p dataframe, where n is the number of rows and p is the number of variables (d < p). To find an interesting structure, a scalar function called the projection pursuit index is optimized. In literature, projection pursuit indices such as those proposed in Friedman (1987); Posse (1995a); Perisic and Posse (2005) among others, were all based on finding the projection that deviated the most from the standard normal distribution. Thus, a goodness of fit test statistic for normality is a plausible projection pursuit index. However, most of these goodness of fit test statistics have limitations, such as not being rotation invariant and exhibiting increased computational complexity in the multivariate case. Hence, more robust goodness of fit test statistics for multivariate data must be considered. In this work, projection pursuit indices based on recent and more robust test statistics for testing normality, namely the Baringhaus, Henze, Epps, and Pulley (BHEP), energy, and Gaussian kernel energy (GKE) test statistics are proposed as much better indices for projection pursuit. The BHEP and the GKE test statistics are dependent on a tuning parameter. Therefore, novel ways for selecting tuning parameters based on cross-validation techniques are presented. Furthermore, the proposed projection pursuit indices are evaluated to determine if they exhibit ideal behaviors of a projection pursuit index and are able to identify interesting hidden structures in both simulated and real datasets.

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