Exact adaptive constant in density estimation

Cristina Butucea

doi:10.1016/S0764-4442(00)80057-9

Exact adaptive constant in density estimation

Cristina Butucea

Humboldt-Universität zu Berlin

Research output: Contribution to journal › Article › peer-review

Abstract

We estimate a probability density f, from n i.i.d. observations, at a fixed point x₀. We suppose that f belongs to a Sobolev class of unknown regularity β. We compute the constant c = c(q, L,β, f(x₀)), associated to the rate of convergence (log n/n) ^{β- 1/2/2β} such that the estimation risk c^-q(log n/n)^{-q β-1/2/2β} E_f\f_n(x₀) - f(x₀)\^q (q > 1) converges to 1, uniformly in f over the class and β in a fixed set, for all estimators f_n (x₀). We construct the adaptive estimation procedure that attains this rate.

Translated title of the contribution	Constante exacte adaptative dans l'estimation de la densité
Original language	English
Pages (from-to)	535-540
Number of pages	6
Journal	Comptes Rendus de l'Academie des Sciences - Series I: Mathematics
Volume	329
Issue number	6
DOIs	https://doi.org/10.1016/S0764-4442(00)80057-9
Publication status	Published - 15 Sept 1999

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title = "Exact adaptive constant in density estimation",

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T1 - Exact adaptive constant in density estimation

AU - Butucea, Cristina

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N2 - We estimate a probability density f, from n i.i.d. observations, at a fixed point x0. We suppose that f belongs to a Sobolev class of unknown regularity β. We compute the constant c = c(q, L,β, f(x0)), associated to the rate of convergence (log n/n) β- 1/2/2β such that the estimation risk c-q(log n/n)-q β-1/2/2β Ef\fn(x0) - f(x0)\q (q > 1) converges to 1, uniformly in f over the class and β in a fixed set, for all estimators fn (x0). We construct the adaptive estimation procedure that attains this rate.

AB - We estimate a probability density f, from n i.i.d. observations, at a fixed point x0. We suppose that f belongs to a Sobolev class of unknown regularity β. We compute the constant c = c(q, L,β, f(x0)), associated to the rate of convergence (log n/n) β- 1/2/2β such that the estimation risk c-q(log n/n)-q β-1/2/2β Ef\fn(x0) - f(x0)\q (q > 1) converges to 1, uniformly in f over the class and β in a fixed set, for all estimators fn (x0). We construct the adaptive estimation procedure that attains this rate.

U2 - 10.1016/S0764-4442(00)80057-9

DO - 10.1016/S0764-4442(00)80057-9

M3 - Article

AN - SCOPUS:0033567766

SN - 0764-4442

VL - 329

SP - 535

EP - 540

JO - Comptes Rendus de l'Academie des Sciences - Series I: Mathematics

JF - Comptes Rendus de l'Academie des Sciences - Series I: Mathematics

IS - 6

ER -

Exact adaptive constant in density estimation

Abstract

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