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2022 Estimation of conditional mean operator under the bandable covariance structure
Kwangmin Lee, Kyoungjae Lee, Jaeyong Lee
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Electron. J. Statist. 16(1): 1253-1302 (2022). DOI: 10.1214/22-EJS1981

Abstract

We consider high-dimensional multivariate linear regression models, where the joint distribution of covariates and response variables is a multivariate normal distribution with a bandable covariance matrix. The main goal of this paper is to estimate the regression coefficient matrix, which is a function of the bandable covariance matrix. Although the tapering estimator of covariance has the minimax optimal convergence rate for the class of bandable covariances, we show that it is sub-optimal for the regression coefficient; that is, a minimax estimator for the class of bandable covariances may not be a minimax estimator for its functionals. We propose the blockwise tapering estimator of the regression coefficient, which has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We also propose a Bayesian procedure called the blockwise tapering post-processed posterior of the regression coefficient and show that the proposed Bayesian procedure has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We show that the proposed methods outperform the existing methods via numerical studies.

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Kwangmin Lee. Kyoungjae Lee. Jaeyong Lee. "Estimation of conditional mean operator under the bandable covariance structure." Electron. J. Statist. 16 (1) 1253 - 1302, 2022. https://doi.org/10.1214/22-EJS1981

Information

Received: 1 May 2021; Published: 2022
First available in Project Euclid: 15 February 2022

Digital Object Identifier: 10.1214/22-EJS1981

Subjects:
Primary: 62C20 , 62H12
Secondary: 62J05

Keywords: Bandable covariance , conditional mean , Covariance estimation , minimax analysis , post-processed posterior

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Vol.16 • No. 1 • 2022
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