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

Abstract

In the Gaussian sequence model $\mathit{Y}=\mathit{\mu }+\mathit{\xi }$, we study the likelihood ratio test (LRT) for testing ${\mathit{H}}_0:\mathit{\mu }={\mathit{\mu }}_0$ versus ${\mathit{H}}_1:\mathit{\mu }\in \mathit{K}$, where ${\mathit{\mu }}_0\in \mathit{K}$, and K is a closed convex set in ${\mathbb{R}}^{\mathit{n}}$. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair $({\mathit{\mu }}_0,\mathit{K})$, 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 nonuniform with respect to the Euclidean metric, and illustrate the conservative nature of existing minimax optimality and suboptimality 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.