• Bernoulli
  • Volume 16, Number 4 (2010), 1137-1163.

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

Evarist Giné and Richard Nickl

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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.

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Bernoulli, Volume 16, Number 4 (2010), 1137-1163.

First available in Project Euclid: 18 November 2010

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adaptive estimation Lepski’s method Rademacher processes spline estimator sup-norm loss wavelet estimator


Giné, Evarist; Nickl, Richard. Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli 16 (2010), no. 4, 1137--1163. doi:10.3150/09-BEJ239.

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