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

2020/10/07 by Han, Qiyang, Sen, Bodhisattva, Shen, Yandi
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2010.03145

Abstract

In the Gaussian sequence model Y=μ+ξ, we study the likelihood ratio test (LRT) for testing H₀: μ=μ₀ versus H₁: μ∈ K, where μ₀ ∈ K, and K is a closed convex set in ℝⁿ. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair (μ₀,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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