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In the Gaussian sequence model $Y=\mu+\xi$, we study the likelihood ratiotest (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, weshow that under the null hypothesis, normal approximation holds for thelog-likelihood ratio statistic for a general pair $(\mu_0,K)$, in the highdimensional regime where the estimation error of the associated least squaresestimator diverges in an appropriate sense. The normal approximation furtherleads to a precise characterization of the power behavior of the LRT in thehigh dimensional regime. These characterizations show that the power behaviorof the LRT is in general non-uniform with respect to the Euclidean metric, andillustrate the conservative nature of existing minimax optimality andsub-optimality results for the LRT. A variety of examples, including testing inthe orthant/circular cone, isotonic regression, Lasso, and testing parametricassumptions versus shape-constrained alternatives, are worked out todemonstrate the versatility of the developed theory.

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