Density estimation has evolved enormously since the days of bar plots and histograms, but researchers and users are still struggling with the problem of the selection of the bin widths. This text explores a new paradigm for the data-based or automatic selection of the free parameters of density estimates in general so that the expected error is within a given constant multiple of the best possible error. The paradigm can be used in nearly all density estimates and for most model selection problems, both parametric and nonparametric. It is the first book on this topic. The text is intended for first-year graduate students in statistics and learning theory, and offers a host of opportunities for further research and thesis topics. Each chapter corresponds roughly to one lecture, and is supplemented with many classroom exercises. A one year course in probability theory at the level of Feller's Volume 1 should be more than adequate preparation. Gabor Lugosi is Professor at Universitat Pompeu Fabra in Barcelona, and Luc Debroye is Professor at McGill University in Montreal. In 1996, the authors, together with Lászlo Györfi, published the successful text, A Probabilistic Theory of Pattern Recognition with Springer-Verlag. Both authors have made many contributions in the area of nonparametric estimation.

Density estimation has evolved enormously since the days of bar plots and histograms, but researchers and users are still struggling with the problem of the selection of the bin widths. This book is the first to explore a new paradigm for the data-based or automatic selection of the free parameters of density estimates in general so that the expected error is within a given constant multiple of the best possible error. The paradigm can be used in nearly all density estimates and for most model selection problems, both parametric and nonparametric.

Users of density estimation methods still struggle with selection of bin widths. This text explores a paradigm for data-based or automatic selection of free parameters of density estimates in general so that expected error is within a given constant multiple of best possible error.

Introduction.- Concentration Inequalities.- Uniform Deviation Inequalities.- Combinatorial Tools.- Total Variation.- Choosing a Density Estimate from a Collection.- Skeleton Estimates.- The Minimum Distance Estimate: Examples.- The Kernel Density Estimate.- Additive Estimates and Data Splitting.- Bandwidth Selection for Kernel Estimates.- Multiparameter Kernel Estimates.- Wavelet Estimates.- The Transformed Kernel Estimate.- Minimax Theory.- Choosing the Kernel Order.- Bandwidth Choice with Superkernels.