## Bernoulli

• 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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 18 November 2010

https://projecteuclid.org/euclid.bj/1290092899

Digital Object Identifier
doi:10.3150/09-BEJ239

Mathematical Reviews number (MathSciNet)
MR2759172

Zentralblatt MATH identifier
1207.62082

#### Citation

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. https://projecteuclid.org/euclid.bj/1290092899

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