Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections
Year: 2010
Type: article
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.
Source: Bernoulli
DOI: 10.3150/09-bej239
Authors: Evarist Giné, Richard Nickl
Institutions: University of Connecticut, University of Cambridge
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2103086317