On estimation of nonsmooth functionals of sparse normal means

O. Collier; L. Comminges; A. B. Tsybakov

doi:10.3150/19-BEJ1180

On estimation of nonsmooth functionals of sparse normal means

O. Collier
, L. Comminges
, A. B. Tsybakov

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

Abstract

We study the problem of estimation of N_γ (θ) = ∑^d_i=₁ | θ_i|^γ for γ > 0 and of the l_γ -norm of θ for γ ≥ 1 based on the observations yi = θ_i + εξ_i, i = 1, . . ., d, where θ = (θ₁, . . ., θ_d) are unknown parameters, ε > 0 is known, and ξ_i are i.i.d. standard normal random variables. We find the non-asymptotic minimax rate for estimation of these functionals on the class of s-sparse vectors θ and we propose estimators achieving this rate.

Original language	English
Pages (from-to)	1989-2020
Number of pages	32
Journal	Bernoulli
Volume	26
Issue number	3
DOIs	https://doi.org/10.3150/19-BEJ1180
Publication status	Published - 1 Aug 2020
Externally published	Yes

Keywords

Functional estimation
Nonsmooth functional
Norm estimation
Polynomial approximation
Sparsity

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title = "On estimation of nonsmooth functionals of sparse normal means",

abstract = "We study the problem of estimation of Nγ (θ) = ∑di=1 | θi|γ for γ > 0 and of the lγ -norm of θ for γ ≥ 1 based on the observations yi = θi + εξi, i = 1, . . ., d, where θ = (θ1, . . ., θd) are unknown parameters, ε > 0 is known, and ξi are i.i.d. standard normal random variables. We find the non-asymptotic minimax rate for estimation of these functionals on the class of s-sparse vectors θ and we propose estimators achieving this rate.",

keywords = "Functional estimation, Nonsmooth functional, Norm estimation, Polynomial approximation, Sparsity",

author = "O. Collier and L. Comminges and Tsybakov, \{A. B.\}",

note = "Publisher Copyright: {\textcopyright} 2020 ISI/BS",

year = "2020",

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doi = "10.3150/19-BEJ1180",

language = "English",

volume = "26",

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TY - JOUR

T1 - On estimation of nonsmooth functionals of sparse normal means

AU - Collier, O.

AU - Comminges, L.

AU - Tsybakov, A. B.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - We study the problem of estimation of Nγ (θ) = ∑di=1 | θi|γ for γ > 0 and of the lγ -norm of θ for γ ≥ 1 based on the observations yi = θi + εξi, i = 1, . . ., d, where θ = (θ1, . . ., θd) are unknown parameters, ε > 0 is known, and ξi are i.i.d. standard normal random variables. We find the non-asymptotic minimax rate for estimation of these functionals on the class of s-sparse vectors θ and we propose estimators achieving this rate.

AB - We study the problem of estimation of Nγ (θ) = ∑di=1 | θi|γ for γ > 0 and of the lγ -norm of θ for γ ≥ 1 based on the observations yi = θi + εξi, i = 1, . . ., d, where θ = (θ1, . . ., θd) are unknown parameters, ε > 0 is known, and ξi are i.i.d. standard normal random variables. We find the non-asymptotic minimax rate for estimation of these functionals on the class of s-sparse vectors θ and we propose estimators achieving this rate.

KW - Functional estimation

KW - Nonsmooth functional

KW - Norm estimation

KW - Polynomial approximation

KW - Sparsity

U2 - 10.3150/19-BEJ1180

DO - 10.3150/19-BEJ1180

M3 - Article

AN - SCOPUS:85091139630

SN - 1350-7265

VL - 26

SP - 1989

EP - 2020

JO - Bernoulli

JF - Bernoulli

IS - 3

ER -

On estimation of nonsmooth functionals of sparse normal means

Abstract

Keywords

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