Available at: https://digitalcommons.calpoly.edu/theses/3295
Date of Award
6-2026
Degree Name
MS in Statistics
Department/Program
Statistics
College
College of Science and Mathematics
Advisor
Trevor Ruiz
Advisor Department
Statistics
Advisor College
College of Science and Mathematics
Abstract
M-estimation provides a unified framework for statistical procedures defined as optimizers of data-dependent criterion functions. This thesis gives an expository account of M-estimation in classical and high-dimensional settings. The classical part develops weak convergence, empirical process tools, and the argmax framework for studying consistency, rates of convergence, and weak limits. Examples including least squares, maximum likelihood, robust location estimation, change-point estimation, and empirical risk minimization illustrate regular and non-regular asymptotic behavior.
The high-dimensional part studies regularized M-estimators, where the focus shifts to finite-sample error bounds and model selection guarantees. Topics include decomposable regularizers, restricted strong convexity, non-convex penalties, and sparsistency. Overall, the thesis emphasizes the common optimization-based structure underlying M-estimation across classical and high-dimensional regimes.