Estimator selection in the gaussian setting

Yannick Baraud; Christophe Giraud; Sylvie Huet

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 1092-1119
  • ISSN: 0246-0203

Abstract

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We consider the problem of estimating the mean f of a Gaussian vector Y with independent components of common unknown variance σ 2 . Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection 𝔽 of estimators of f based on Y and, with the same data Y , aim at selecting an estimator among 𝔽 with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to Y may be unknown. We establish a non-asymptotic risk bound for the selected estimator and derive oracle-type inequalities when 𝔽 consists of linear estimators. As particular cases, our approach allows to handle the problems of aggregation, model selection as well as those of choosing a window and a kernel for estimating a regression function, or tuning the parameter involved in a penalized criterion. In all theses cases but aggregation, the method can be easily implemented. For illustration, we carry out two simulation studies. One aims at comparing our procedure to cross-validation for choosing a tuning parameter. The other shows how to implement our approach to solve the problem of variable selection in practice.

How to cite

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Baraud, Yannick, Giraud, Christophe, and Huet, Sylvie. "Estimator selection in the gaussian setting." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 1092-1119. <http://eudml.org/doc/271982>.

@article{Baraud2014,
abstract = {We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance $\sigma ^\{2\}$. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $\mathbb \{F\}$ of estimators of $f$ based on $Y$ and, with the same data $Y$, aim at selecting an estimator among $\mathbb \{F\}$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown. We establish a non-asymptotic risk bound for the selected estimator and derive oracle-type inequalities when $\mathbb \{F\}$ consists of linear estimators. As particular cases, our approach allows to handle the problems of aggregation, model selection as well as those of choosing a window and a kernel for estimating a regression function, or tuning the parameter involved in a penalized criterion. In all theses cases but aggregation, the method can be easily implemented. For illustration, we carry out two simulation studies. One aims at comparing our procedure to cross-validation for choosing a tuning parameter. The other shows how to implement our approach to solve the problem of variable selection in practice.},
author = {Baraud, Yannick, Giraud, Christophe, Huet, Sylvie},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {estimator selection; model selection; variable selection; linear estimator; kernel estimator; ridge regression; Lasso; elastic net; random forest; PLS1 regression; lasso},
language = {eng},
number = {3},
pages = {1092-1119},
publisher = {Gauthier-Villars},
title = {Estimator selection in the gaussian setting},
url = {http://eudml.org/doc/271982},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Baraud, Yannick
AU - Giraud, Christophe
AU - Huet, Sylvie
TI - Estimator selection in the gaussian setting
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 1092
EP - 1119
AB - We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance $\sigma ^{2}$. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $\mathbb {F}$ of estimators of $f$ based on $Y$ and, with the same data $Y$, aim at selecting an estimator among $\mathbb {F}$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown. We establish a non-asymptotic risk bound for the selected estimator and derive oracle-type inequalities when $\mathbb {F}$ consists of linear estimators. As particular cases, our approach allows to handle the problems of aggregation, model selection as well as those of choosing a window and a kernel for estimating a regression function, or tuning the parameter involved in a penalized criterion. In all theses cases but aggregation, the method can be easily implemented. For illustration, we carry out two simulation studies. One aims at comparing our procedure to cross-validation for choosing a tuning parameter. The other shows how to implement our approach to solve the problem of variable selection in practice.
LA - eng
KW - estimator selection; model selection; variable selection; linear estimator; kernel estimator; ridge regression; Lasso; elastic net; random forest; PLS1 regression; lasso
UR - http://eudml.org/doc/271982
ER -

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