Structural adaptation via double-struck Lp-norm oracle inequalities

Alexander Goldenshluger; Oleg Lepski

doi:10.1007/s00440-007-0119-5

Structural adaptation via double-struck L_p-norm oracle inequalities

Alexander Goldenshluger, Oleg Lepski

Department of Statistics

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

Abstract

In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary double-struck L_p-losses. These results are applied for adaptive estimation in the additive multi-index model.

Original language	English
Pages (from-to)	41-71
Number of pages	31
Journal	Probability Theory and Related Fields
Volume	143
Issue number	1-2
DOIs	https://doi.org/10.1007/s00440-007-0119-5
State	Published - Jan 2009

Bibliographical note

Funding Information:
Supported by the ISF grant No. 389/07.

Keywords

Adaptive estimation
Minimax risk
Optimal rates of convergence
Oracle inequalities
Structural adaptation

ASJC Scopus subject areas

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1007/s00440-007-0119-5

Cite this

@article{027459bb14b44b889dd2ac2fd669c80d,

title = "Structural adaptation via double-struck Lp-norm oracle inequalities",

abstract = "In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary double-struck Lp-losses. These results are applied for adaptive estimation in the additive multi-index model.",

keywords = "Adaptive estimation, Minimax risk, Optimal rates of convergence, Oracle inequalities, Structural adaptation",

author = "Alexander Goldenshluger and Oleg Lepski",

note = "Funding Information: Supported by the ISF grant No. 389/07.",

year = "2009",

month = jan,

doi = "10.1007/s00440-007-0119-5",

language = "English",

volume = "143",

pages = "41--71",

journal = "Probability Theory and Related Fields",

issn = "0178-8051",

publisher = "Springer New York",

number = "1-2",

}

TY - JOUR

T1 - Structural adaptation via double-struck Lp-norm oracle inequalities

AU - Goldenshluger, Alexander

AU - Lepski, Oleg

N1 - Funding Information: Supported by the ISF grant No. 389/07.

PY - 2009/1

Y1 - 2009/1

N2 - In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary double-struck Lp-losses. These results are applied for adaptive estimation in the additive multi-index model.

AB - In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary double-struck Lp-losses. These results are applied for adaptive estimation in the additive multi-index model.

KW - Adaptive estimation

KW - Minimax risk

KW - Optimal rates of convergence

KW - Oracle inequalities

KW - Structural adaptation

UR - http://www.scopus.com/inward/record.url?scp=54349118425&partnerID=8YFLogxK

U2 - 10.1007/s00440-007-0119-5

DO - 10.1007/s00440-007-0119-5

M3 - Article

AN - SCOPUS:54349118425

SN - 0178-8051

VL - 143

SP - 41

EP - 71

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -