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 $\mathbb R$ 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 H\"{o}lder balls, also in sup-norm loss. The estimators are obtained from 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. Explicit constants in the asymptotic risk of the estimator are obtained, as well as oracle-type inequalities in sup-norm loss. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-analogues of the inequalities in Koltchinskii (2006) for the deviation of suprema of empirical processes from their Rademacher symmetrizations.