  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.
