Statistics Papers

Document Type

Journal Article

Date of this Version

2-2016

Publication Source

The Annals of Statistics

Abstract

This paper considers point and interval estimation of the ℓq loss of an estimator in high-dimensional linear regression with random design. We establish the minimax rate for estimating the ℓq loss and the minimax expected length of confidence intervals for the ℓq loss of rate-optimal estimators of the regression vector, including commonly used estimators such as Lasso, scaled Lasso, square-root Lasso and Dantzig Selector. Adaptivity of the confidence intervals for the ℓq loss is also studied. Both the setting of known identity design covariance matrix and known noise level and the setting of unknown design covariance matrix and unknown noise level are studied. The results reveal interesting and significant differences between estimating the ℓ2 loss and ℓq loss with 1 ≤ q < 2 as well as between the two settings. New technical tools are developed to establish rate sharp lower bounds for the minimax estimation error and the expected length of minimax and adaptive confidence intervals for the ℓq loss. A significant difference between loss estimation and the traditional parameter estimation is that for loss estimation the constraint is on the performance of the estimator of the regression vector, but the lower bounds are on the difficulty of estimating its ℓq loss. The technical tools developed in this paper can also be of independent interest.

Keywords

Accuracy assessment, adaptivity, confidence interval, highdimensional linear regression, loss estimation, minimax lower bound, minimaxity, sparsity

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Date Posted: 27 November 2017

This document has been peer reviewed.