# Robust High-dimensional Tuning Free Multiple Testing

arXiv: https://arxiv.org/abs/2211.11959 · HTML: https://mlsanity.com/paper/2211.11959/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 93 references that are in the map.
- First posted on arXiv 22 Nov 2022.

## Builds on (most cited first, 18 of 18)

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [Loss minimization and parameter estimation with heavy tails](https://mlsanity.com/paper/1307.1827/) (2013)
- [Empirical risk minimization for heavy-tailed losses](https://mlsanity.com/paper/1406.2462/) (2014)
- [Statistical consistency and asymptotic normality for high-dimensional robust M-estimators](https://mlsanity.com/paper/1501.00312/) (2015)
- [How do noise tails impact on deep ReLU networks?](https://mlsanity.com/paper/2203.10418/) (2022)
- [A Shrinkage Principle for Heavy-Tailed Data: High-Dimensional Robust Low-Rank Matrix Recovery](https://mlsanity.com/paper/1603.08315/) (2016)
- [Can parametric statistical methods be trusted for fMRI based group studies?](https://mlsanity.com/paper/1511.01863/) (2015)
- [Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries](https://mlsanity.com/paper/1605.07129/) (2016)
- [Understanding Implicit Regularization in Over-Parameterized Single Index Model](https://mlsanity.com/paper/2007.08322/) (2020)
- [Robust Inference via Multiplier Bootstrap](https://mlsanity.com/paper/1903.07208/) (2019)
- [FarmTest: Factor-Adjusted Robust Multiple Testing with Approximate False Discovery Control](https://mlsanity.com/paper/1711.05386/) (2017)
- [Globally adaptive quantile regression with ultra-high dimensional data](https://mlsanity.com/paper/1507.00420/) (2015)
- [A New Perspective on Robust M-Estimation: Finite Sample Theory and Applications to Dependence-Adjusted Multiple Testing](https://mlsanity.com/paper/1711.05381/) (2017)
- [Are Latent Factor Regression and Sparse Regression Adequate?](https://mlsanity.com/paper/2203.01219/) (2022)
- [Robust Matrix Completion with Heavy-tailed Noise](https://mlsanity.com/paper/2206.04276/) (2022)
- [Phase Transition and Regularized Bootstrap in Large Scale t-tests with False Discovery Rate Control](https://mlsanity.com/paper/1310.4371/) (2013)
- [Simulation-Based Hypothesis Testing of High Dimensional Means Under Covariance Heterogeneity](https://mlsanity.com/paper/1406.1939/) (2014)

## Abstract (arXiv)

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'er 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 tuning-free and 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.