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

By Evarist Gin'e, Richard Nickl. 2008

34 citations recorded by the source corpus.

Given an i.i.d. sample from a distribution F on R with uniformly continuous density p0, purely-data driven estimators are constructed that eciently estimate F in sup-norm loss, and simultaneously estimate p0 at the best possible rate of convergence over Holder 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. 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. MSC 2000 subject classication : Primary: 62G07; Secondary: 60F05.

Subfields: Applied Mathematics, Control and Systems Engineering, Statistics and Probability

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Citations in the corpus

On the optimal estimation of probability measures in weak and strong topologies (2013, background): “Several recent works [6, 26, 16, 17, 18, 19] have shown that many popular density estimators on X = R, such as maximum likelihood estimator, kernel density estimator and wavelet estimator satisfy (2) if F is P-Donsker—the Donsker classes that were considered in these works are: functions of bounded ”
Anti-concentration and honest, adaptive confidence bands (2013, methodology): “Previous literature on the Lepski method used Talagrand’s inequality combined with some bounds on expected suprema of certain empirical processes (obtained via symmetrization and entropy methods) to choose the threshold level for the estimator [the right-hand side of the inequality in (35)]; see [19”
Adaptive density estimation: a curse of support? ✩ (2009, unknown): “Note that the case p ′ = ∞ is very di ﬀ erent in nature (see Gin´e and Nickl (2009) and Gin´e and Nickl (2010)).”
CONFIDENCE BANDS IN DENSITY ESTIMATION (2010, unknown): “48 01 v1 [ m at h. ST ] 2 5 Fe b 20 10 The Annals of Statistics 2010, Vol. 38, No. 2, 1122–1170 DOI: 10.1214/09-AOS738 c© Institute of Mathematical Statistics, 2010”
Oracle inequalities in empirical risk minimization and sparse recovery problems (2011, background): “Recently, Giné and Nickl [51] used some inequalities of similar nature in adaptive density estimation.”
On adaptive inference and confidence bands (2011, background): “About question (ii): if the radius B of the Hölder balls involved is known, then one can combine a natural testing approach with recent results in [10, 11, 13] to prove the existence of adaptive nonparametric confidence bands over a scale of Hölder classes indexed by a grid of smoothness parameters ”