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November 2010 Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections
Evarist Giné, Richard Nickl
Bernoulli 16(4): 1137-1163 (November 2010). DOI: 10.3150/09-BEJ239

Abstract

Given an i.i.d. sample from a distribution $F$ on $ℝ$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ 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.

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Evarist Giné. Richard Nickl. "Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections." Bernoulli 16 (4) 1137 - 1163, November 2010. https://doi.org/10.3150/09-BEJ239

Information

Published: November 2010
First available in Project Euclid: 18 November 2010

zbMATH: 1207.62082
MathSciNet: MR2759172
Digital Object Identifier: 10.3150/09-BEJ239

Keywords: adaptive estimation , Lepski’s method , Rademacher processes , spline estimator , sup-norm loss , wavelet estimator

Rights: Copyright © 2010 Bernoulli Society for Mathematical Statistics and Probability

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Vol.16 • No. 4 • November 2010
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