Maximum penalized likelihood estimation - Volume 1 : Density Estimation

This text deals with parametric and nonparametric density estimation from the maximum (penalized) likelihood point of view, including estimation under constraints such as unimodality and log-concavity. It is intended for graduate students in statistics, applied mathematics, and operations research, as well as for researchers and practitioners in the field.
The focal points are existence and uniqueness of the estimators, almost sure convergence rates for the L1 error, and data-driven smoothing parameter selection methods, including their practical performance. The reader will gain insight into some of the generally applicable technical tools from probability theory (discrete parameter martingales) and applied mathematics (boundary, value problems and integration by parts tricks.) Convexity and convex optimization, as applied to maximum penalized likelihood estimation, receive special attention.
The authors are with the Statistics Program of the Department of Food and Resource Economics in the College of Agriculture at the University of Delaware.

Contents

• Parametric Maximum Likelihood Estimation
• Parametric Maximum Likelihood Estimation in Action
• Kernel Density Estimation
• Maximum Likelihood Density Estimation
• Monotone and Unimodal Densities
• Choosing the Smoothing Parameter
• Nonparametric Density Estimation in Action
• Convex Minimization in Finite Dimensional Spaces
• Convex Minimization in Infinite Dimensional Spaces
• Convexity in Action