Optimal prediction for linear regression with infinitely many parameters

Alexander Goldenshluger; Alexandre Tsybakov

doi:10.1016/S0047-259X(02)00006-4

Optimal prediction for linear regression with infinitely many parameters

Alexander Goldenshluger, Alexandre Tsybakov

Department of Statistics

Research output: Contribution to journal › Review article › peer-review

Abstract

The problem of optimal prediction in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that outperforms asymptotically the ordinary least squares predictor. Moreover, if the random errors are Gaussian, the method is asymptotically minimax over ellipsoids in ℓ₂. The method is based on a regularized least squares estimator with weights of the Pinsker filter. We also consider the case of dynamic linear regression, which is important in the context of transfer function modeling.

Original language	English
Pages (from-to)	40-60
Number of pages	21
Journal	Journal of Multivariate Analysis
Volume	84
Issue number	1
DOIs	https://doi.org/10.1016/S0047-259X(02)00006-4
State	Published - Jan 2003

Bibliographical note

Funding Information:
The research was supported in part by a Grant of the ESF programme on “Highly Structured Stochastic Systems (HSSS)” and by a Grant of the Arc-en-ciel/Keshet program.

Keywords

Exact asymptotics of minimax risk
Linear regression with infinitely many parameters
Optimal prediction
Pinsker filter

ASJC Scopus subject areas

Statistics and Probability
Numerical Analysis
Statistics, Probability and Uncertainty

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10.1016/S0047-259X(02)00006-4

Cite this

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title = "Optimal prediction for linear regression with infinitely many parameters",

abstract = "The problem of optimal prediction in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that outperforms asymptotically the ordinary least squares predictor. Moreover, if the random errors are Gaussian, the method is asymptotically minimax over ellipsoids in ℓ2. The method is based on a regularized least squares estimator with weights of the Pinsker filter. We also consider the case of dynamic linear regression, which is important in the context of transfer function modeling.",

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author = "Alexander Goldenshluger and Alexandre Tsybakov",

note = "Funding Information: The research was supported in part by a Grant of the ESF programme on “Highly Structured Stochastic Systems (HSSS)” and by a Grant of the Arc-en-ciel/Keshet program.",

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AU - Tsybakov, Alexandre

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N2 - The problem of optimal prediction in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that outperforms asymptotically the ordinary least squares predictor. Moreover, if the random errors are Gaussian, the method is asymptotically minimax over ellipsoids in ℓ2. The method is based on a regularized least squares estimator with weights of the Pinsker filter. We also consider the case of dynamic linear regression, which is important in the context of transfer function modeling.

AB - The problem of optimal prediction in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that outperforms asymptotically the ordinary least squares predictor. Moreover, if the random errors are Gaussian, the method is asymptotically minimax over ellipsoids in ℓ2. The method is based on a regularized least squares estimator with weights of the Pinsker filter. We also consider the case of dynamic linear regression, which is important in the context of transfer function modeling.

KW - Exact asymptotics of minimax risk

KW - Linear regression with infinitely many parameters

KW - Optimal prediction

KW - Pinsker filter

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U2 - 10.1016/S0047-259X(02)00006-4

DO - 10.1016/S0047-259X(02)00006-4

M3 - Review article

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SN - 0047-259X

VL - 84

SP - 40

EP - 60

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

IS - 1

ER -

Optimal prediction for linear regression with infinitely many parameters

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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