High dimensional asymptotics of likelihood ratio tests in the Gaussian sequence model under convex constraints

TL;DR

This paper analyzes the likelihood ratio test in high-dimensional Gaussian models with convex constraints, revealing normal approximation validity and detailed power behavior, with applications to various statistical testing scenarios.

Contribution

It provides a high-dimensional asymptotic analysis of the LRT under convex constraints, including normal approximation and power characterization, extending existing theory to complex models.

Findings

Normal approximation holds for the LRT in high dimensions.

LRT power behavior is non-uniform and context-dependent.

The theory applies to diverse models like Lasso and shape constraints.

Abstract

In the Gaussian sequence model $Y=\mu+\xi$, we study the likelihood ratio test (LRT) for testing $H_0: \mu=\mu_0$ versus $H_1: \mu \in K$, where $\mu_0 \in K$, and $K$ is a closed convex set in $\mathbb{R}^n$. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair $(\mu_0,K)$, in the high dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense. The normal approximation further leads to a precise characterization of the power behavior of the LRT in the high dimensional regime. These characterizations show that the power behavior of the LRT is in general non-uniform with respect to the Euclidean metric, and illustrate the conservative nature of existing minimax optimality and sub-optimality results for the LRT. A variety of examples, including testing in the orthant/circular cone, isotonic regression, Lasso, and testing parametric assumptions versus shape-constrained alternatives, are worked out to demonstrate the versatility of the developed theory.