Combinatorial Methods in Density Estimation

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Springer Science & Business Media, 2012年12月6日 - 209 頁
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.
 

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內容

Introduction
1
Uniform Deviation Inequalities
17
Combinatorial Tools
27
13
36
Total Variation
38
Choosing a Density Estimate
47
Skeleton Estimates
58
Examples
70
The Kernel Density Estimate
79
Minimax Theory
150
Choosing the Kernel Order
177
Bandwidth Choice with Superkernels
190
Author Index
199
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