Maximum Penalized Likelihood Estimation, Volume I, Density Estimation
Pia Veldt Larsen · Journal of the Royal Statistical Society Series A (Statistics in Society) · 2004
This is the first part of a proposed two-volume text on maximum likelihood estimation. This volume treats density estimation, whereas volume II will treat indirect estimation problems. The emphasis is on nonparametric density estimation. Topics of particular interest are the existence and uniqueness of estimators, almost sure convergence, rates of the L1-error and selection methods for smoothing parameters. Convexity and convex optimization applied to maximum penalized likelihood estimation are discussed in detail. Throughout the book, applications and the prac- tical performance of the theoretical results are studied. Chapter 1 presents a general overview of parametric and nonparametric estimation. The remaining chapters split into three parts: part one (Chapters 2 and 3) reviews parametric density estimation, part two (Chapters 4–8) discusses nonparametric density estimation and, finally, part three (Chapters 9–11) is concerned with convexity and optimization. Each part consists of one or more theoretical chapters and an ‘In-action’ chapter, in which the theory is considered from a practical point of view. Part one looks at parametric maximum likelihood estimation, both theory and practice. It discusses the EM algorithm, M-estimators, ridge regression and heavy tail regression. Part two discusses kernel density estimation, nonparametric maximum penalized likelihood estimation (the authors are particularly fond of Good's roughness penalization), monotone and unimodal densities and the selection of smoothing parameters. The in-action Chapter 8 examines nonparametric density estimation. In part three, convex optimization in finite dimensional spaces and in infinite dimensional spaces respectively are discussed in Chapters 9 and 10. Chapter 11 considers convexity from a practical point of view. The book provides a good and up-to-date introduction to nonparametric density estimation. One of its main strengths is giving overviews and motivations of the general ideas before moving on to the technicalities. This, together with the in-action chapters, makes it an excellent text-book for grad- uate students in statistics, as well as practition- ers in the field. At times, the notation is a little difficult but once one gets used to it (the list of notation, acronyms and conventions is very useful) the mathematical level is no higher than necessary.