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 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.