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

In the Gaussian sequence model Y=?+?, we study likelihood ratio test (LRT) for testing H0:?=?0 versus H1:??K, where ?0?K, and K is a closed convex set in Rn. particular, show that under null hypothesis, normal approximation holds log-likelihood statistic general pair (?0,K), high-dimensional regime estimation error of associated least squares estimator diverges an appropriate sense. The further leads to precise characterization power behavior LRT regime. These characterizations nonuniform with respect Euclidean metric, illustrate conservative nature existing minimax optimality suboptimality results LRT. A variety examples, including orthant/circular cone, isotonic regression, Lasso parametric assumptions shape-constrained alternatives, are worked out demonstrate versatility developed theory.