论文标题
高斯序列模型中的高维渐近学测试在凸约束下
High dimensional asymptotics of likelihood ratio tests in the Gaussian sequence model under convex constraints
论文作者
论文摘要
在高斯序列模型中，$ y =μ+ξ$，我们研究了用于测试$ H_0的可能性比测试（LRT）：μ=μ_0$ vess $ h_1：μ\ in K $，其中$μ_0\，其中$μ_0\ in k $ in K $，$ k $是$ k $是$ \ mathbbbbbb^n $ n $ \ mathbb^n $。特别是，我们表明，在零假设下，在高维度方向上，对数可能的比例比（μ_0，k）$的对数似然比统计量的正常近似值，在适当的含义上，相关最小二乘估计值的估计误差在高维状态下。正常近似进一步导致了在高维状态下LRT功率行为的精确表征。这些特征表明，相对于欧几里得指标，LRT的功率行为通常是不均匀的，并说明了LRT的现有最小值优化性和亚次优势的保守性质。制定了各种示例，包括在矫形/圆锥锥中进行测试，等渗回归，LASSO和测试参数假设与形状约束的替代方案，以证明已发达理论的多功能性。
In the Gaussian sequence model $Y=μ+ξ$, we study the likelihood ratio test (LRT) for testing $H_0: μ=μ_0$ versus $H_1: μ\in K$, where $μ_0 \in K$, 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 $(μ_0,K)$, 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.
