Robust High-dimensional Tuning Free Multiple Testing

• URL: http://arxiv.org/abs/2211.11959v2
• Date: Wed, 23 Nov 2022 18:20:18 GMT
• Title: Robust High-dimensional Tuning Free Multiple Testing
• Authors: Jianqing Fan, Zhipeng Lou, Mengxin Yu
• Abstract summary: This paper revisits the celebrated Hodges-Lehmann (HL) estimator for estimating location parameters in both the one- and two-sample problems. We develop Berry-Esseen inequality and Cram'er type moderate deviation for the HL estimator based on newly developed non-asymptotic Bahadur representation. It is convincingly shown that the resulting tuning-free and moment-free methods control false discovery proportion at a prescribed level.
• Score: 0.49416305961918056
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A stylized feature of high-dimensional data is that many variables have heavy tails, and robust statistical inference is critical for valid large-scale statistical inference. Yet, the existing developments such as Winsorization, Huberization and median of means require the bounded second moments and involve variable-dependent tuning parameters, which hamper their fidelity in applications to large-scale problems. To liberate these constraints, this paper revisits the celebrated Hodges-Lehmann (HL) estimator for estimating location parameters in both the one- and two-sample problems, from a non-asymptotic perspective. Our study develops Berry-Esseen inequality and Cram\'{e}r type moderate deviation for the HL estimator based on newly developed non-asymptotic Bahadur representation, and builds data-driven confidence intervals via a weighted bootstrap approach. These results allow us to extend the HL estimator to large-scale studies and propose \emph{tuning-free} and \emph{moment-free} high-dimensional inference procedures for testing global null and for large-scale multiple testing with false discovery proportion control. It is convincingly shown that the resulting tuning-free and moment-free methods control false discovery proportion at a prescribed level. The simulation studies lend further support to our developed theory.

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