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

Abstract

Given an i.i.d. sample from a distribution $F$ on $\mathbb{R}$ 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.

Published in at http://dx.doi.org/10.3150/09-BEJ239 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (5)

• 2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization
• Concentration around the mean for maxima of empirical processes
• Concentration inequalities and asymptotic results for ratio type empirical processes
• Uniform limit theorems for wavelet density estimators
• Structural adaptation via $L_p$-norm oracle inequalities

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• Adaptive pointwise estimation of conditional density function
• Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme
• Pliable rejection sampling
• Improved Rademacher symmetrization through a Wasserstein based measure of asymmetry