Nonparametric density estimation. The \(L_ 1\) view. (English) Zbl 0546.62015

Devroye, Luc; Györfi, László

Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons. XI, 356 p. £45.90 (1985).

Treatment of nonparametric density estimation has been scattered in a few specialized monographs, general statistics texts and in approximately 1000 research papers. This monograph offers a single-course examination of the subject. It develops, from first principles, the ”natural” theory for density estimation, \(L_ 1\), and shows why the classical \(L_ 2\)- theory masks some fundamental properties of density estimates. Linking together different subareas of statistics, including simulations, pattern recognition, detection theory and minimax theory, it shows how to construct, use and analyze density estimates. Additionally, the relevant recent literature is tied in with the more classical works of Parzen, Rosenblatt and others.
The chapters comprehensively examine consistency, lower bounds for rates of convergence, rates of convergence in \(L_ 1\), the transformed kernel estimate, applications in discriminant analysis and estimators based on orthogonal series. A detailed keyword and author index and current bibliography complement the book.
With applications as diverse as improving the performance of signal detectors and pattern recognition machines or sizing statistical samples for engineering, this book provides statisticians, engineers and researchers with an invaluable reference guide. Professors and advanced graduate students in probability, statistics, computer science and electrical engineering can increase their understanding by following the \(L_ 1\) approach.

Reviewer: J.Antoch

MSC:

62G05	Nonparametric estimation
62-02	Research exposition (monographs, survey articles) pertaining to statistics
62P99	Applications of statistics
62N99	Survival analysis and censored data

Keywords:

\(L_1\) approach; orthogonal series estimators; density estimation; simulations; pattern recognition; detection; consistency; bounds; rates of convergence; transformed kernel estimate; discriminant analysis