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.
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Uniform Deviation Inequalities
Choosing a Density Estimate
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Annals of Statistics Assume asymptotic Bibliographic Remarks Birgé Borel sets Cauchy–Schwarz inequality cells Chapter Chervonenkis class of densities class of sets combinatorial defined denote densi density estimate density f Devroye empirical measure esti estimate fri example finite fixed func Györfi Hellinger distance histogram inequality integrable intervals Jensen's inequality kernel complexity Kolmogorov entropy L1 distance L1 error Lebesgue LEBESGUE DENSITY THEOREM Lemma Let f Lipschitz log-concave lower bound Lugosi mate minimax minimum distance estimate monotone multivariate nonnegative nonparametric normal density number of different obtain optimal parameter partition polynomial PROOF random variables rate of convergence real line Riemann shatter coefficient Show skeleton estimate ſlf ſº ſº-ſº Talagrand Theorem Theory timate tion total variation total variation distance transformed un(A uniform unimodal upper bound Vapnik Vapnik–Chervonenkis dimension variable kernel estimate vector wavelet Yatracos class zero