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

时间:2021-09-06         阅读:

光华讲坛——社会名流与企业家论坛第5858期

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

主讲人罗格斯大学 韩启阳助理教授

主持人统计学院 常晋源教授

时间2021年9月10日(周五)上午10:00-11:00

举办地点:腾讯会议,662 169 903

主办单位:数据科学与商业智能联合实验室 统计学院 科研处

主讲人简介:

Qiyang Han is an assistant professor of Statistics at Rutgers University. He received a Ph.D. in Statistics from University of Washington under the supervision of Professor Jon A. Wellner. He is broadly interested in mathematical statistics and high dimensional probability, with a particular focus on empirical process theory and its applications to nonparametric and high dimensional statistics.

韩启阳,罗格斯大学统计系助理教授。他博士毕业于华盛顿大学统计学专业,导师是Jon A. Wellner教授。他对数理统计和高维概率有着广泛兴趣,特别关注经验过程理论及其在非参数和高维统计中应用的相关研究。

内容简介

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 $\R^n$. 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.

This talk is based on joint work with Yandi Shen(UW, Chicago) and Bodhisattva Sen(Columbia).

在高斯序列模型$Y=\mu+\xi$中,本文研究了似然比检验(LRT)来检测$H_0:\mu=\mu_0$ vs $H_1:\mu\in K$,其中$\mu_0在K$中,$K$是在$\R^n$中的一个闭凸集。特别是,本文研究表明,在零假设下,在相关最小二乘估计量的估计误差有一定发散的高维情形下,对于$(\mu_0,K)$的对数似然比统计量的正态近似成立。正态近似能进一步获得在高维情形下LRT功效行为的精确特征。这些特征表明,LRT的功效行为相对于欧几里得度量通常是不一致的,并且说明了LRT现有的极小极大优化和次优结果的保守性质。各种例子如在正态/圆锥体中的测试、等渗回归、套索和检验参数假设与形状约束等都可证明本文理论的适用性。

本文是和Yandi Shen(华盛顿,芝加哥)、Bodhisattva Sen(哥伦比亚)合作研究的。

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