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.
@article{1290092899,
author = {Gin\'e, Evarist and Nickl, Richard},
title = {Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections},
journal = {Bernoulli},
volume = {16},
number = {1},
year = {2010},
pages = { 1137-1163},
language = {en},
url = {http://dml.mathdoc.fr/item/1290092899}
}
Giné, Evarist; Nickl, Richard. Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli, Tome 16 (2010) no. 1, pp. 1137-1163. http://gdmltest.u-ga.fr/item/1290092899/