High dimensional asymptotics of likelihood ratio tests in Gaussian sequence model under convex constraint
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In the Gaussian sequence model Y=μ+ξ, we study the likelihood ratio test (LRT) for testing H_0: μ=μ_0 versus H_1: μ∈ K, where μ_0 ∈ K, and K is a closed convex set in ℝ^n. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair (μ_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.
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