Date of Award

2024-08-01

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Advisor(s)

Abhijit A. Mandal

Second Advisor

Michael M. Pokojovy

Abstract

The estimation of the location vector and scatter matrix plays a crucial role in many multivariate statistical methods. However, the classical likelihood-based estimation is greatly influenced by outliers, potentially leading to unreliable decisions. Hence, a fundamental challenge in multivariate statistics is to develop robust alternatives that can maintain performancein the presence of outliers and deviations from the assumed data distribution. Unfortunately, methods with good global robustness often substantially sacrifice efficiency. To address this, we propose the adoption of Minimum Density Power Divergence (MDPD) estimation, a well-established robust technique known for its efficiency and statistical robustness to outliers and model violations. Focusing on multivariate contaminated Gaussian models, we present the first-order optimality conditions associated with minimizing the loss function underlying the MDPD estimator. We also describe a computationally efficient iterative algorithm designed to converge to a local minimum of the loss function. Additionally, we develop a robust one-way Multivariate Analysis of Variance (MANOVA) test based on the MDPD estimator, which is particularly useful for analyzing multiple dependent variables simultaneously, especially when a significant correlation between dependent variables exists. The asymptotic properties of the MDPD estimator and proposed MANOVA test are derived under suitable regularity conditions. Extensive Monte Carlo simulations are further conducted to empirically evaluate the statistical efficiency and quantitative robustness of the proposed methods. Furthermore, we provide real-world dataset examples on robust principal component analysis (PCA), multivariate regression, and one-way MANOVA tests. Our proposed approach is observed to be competitive or superior when compared both to classical likelihood-based methods and robust techniques based on the Minimum Covariance Determinant (MCD) estimator.

Language

en

Provenance

Received from ProQuest

File Size

159 p.

File Format

application/pdf

Rights Holder

Ebenezer Nkum

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