Combinatorial Methods in Density 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 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
1 online resource (xii, 208 pages)
9781461301257, 1461301254
852792053
Print version:
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
English