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

DOI10.1214/21-AOS2111arXiv2010.03145MaRDI QIDQ6350716

Yandi Shen, Qiyang Han, Bodhisattva Sen

Publication date: 7 October 2020

Abstract: In the Gaussian sequence model $Y=mu+xi$, we study the likelihood ratio test (LRT) for testing $H_0:mu=m{u}_0$ versus $H_1:muinK$, where $m{u}_{0}inK$, 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 $(m{u}_0,K)$, 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.

Mathematics Subject Classification ID

Approximations to statistical distributions (nonasymptotic) (62E17) Functional limit theorems; invariance principles (60F17)

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