题目：Beyond Detection Boundary: Minimax Deficiency for Two-Sample Mean Tests in High Dimensions

报告人: 邱宇谋 研究员（北京大学）

地点: 致远楼108室

时间: 2024年10月25日 10:00-11:00

Abstract：The detection boundary is an effective tool for evaluating a high dimensional test procedure. However, it is not a comprehensive performance measure as it is built upon trivial cases where the sum of type I and type II error rates converges to zero or one, which cannot distinguish between the L2 and higher criticism (HC) tests under dense signals nor between the maximum and HC tests under highly sparse signals. To overcome the limitation of the detection boundary, we investigate the nontrivial limits of the minimax type II error rate under the controlled type I error rate for testing two-sample means against a sequence of two-sided local alternatives, and prove a one-to-one correspondence between the signal strength and the type II error rate in a non-asymptotic framework for Gaussian data. Based on these results, we propose two sharper minimax deficiency measures, the minimax relative deficiency and the minimax absolute deficiency, to quantify the differences in the signal strength such that a high dimensional test and the minimax optimal test could share the same nontrivial type I and type II error rates. Those two measures can recover the higher order terms not shown in the detection boundary. Using the proposed minimax deficiency measures, we provide a full evaluation of three basic high dimensional tests for two-sample means and respectively show the superiority of the L2, HC and maximum tests under the dense, moderately sparse and highly sparse signal regimes. To guarantee the robustness against the unknown signal sparsity in practice, we further propose a novel adaptive testing procedure by combining the three tests, which is optimal in terms of the minimax relative deficiency over the whole signal sparsity regime. Simulation studies are conducted to evaluate the proposed test and demonstrate its superiority.

报告人简介: 邱宇谋，博士毕业于爱荷华州立大学，先后在内布拉斯加林肯大学和爱荷华州立大学任教。于2023年7月加入北京大学数学科学学院、统计科学中心，职位为研究员。他的研究包括：高维数据分析、高维协方差矩阵和精度矩阵的统计推断、因果分析、缺失数据分析。同时，他也致力于统计方法在精准农业、流行病模型、法医学等领域的应用研究。

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