Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

Given an i.i.d. sample from a distribution F on ℝ with uniformly continuous density p₀, purely data-driven estimators are constructed that efficiently estimate F in sup-norm loss and simultaneously estimate p₀ at the best possible rate of convergence over Hölder balls, also in sup-norm loss. The est...

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Veröffentlicht in:Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability 2010-11, Vol.16 (4), p.1137-1163
Hauptverfasser: GINÉ, EVARIST, NICKL, RICHARD
Format: Artikel
Sprache:eng
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Zusammenfassung:Given an i.i.d. sample from a distribution F on ℝ with uniformly continuous density p₀, purely data-driven estimators are constructed that efficiently estimate F in sup-norm loss and simultaneously estimate p₀ at the best possible rate of convergence over Hölder balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or B-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593—2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.
ISSN:1350-7265
DOI:10.3150/09-bej239