Estimating minimum effect with outlier selection

TL;DR

This paper develops methods for estimating the minimum effect and identifying outliers in a one-sided contamination model, providing minimax risks, adaptive estimators, and FDR control under dependence.

Contribution

It introduces one-sided contamination models, derives minimax risks, proposes adaptive estimators, and addresses FDR control under correlation, advancing outlier detection and robust mean estimation.

Findings

Optimal convergence rates differ from classical models.

Adaptive estimators perform well with unknown contamination levels.

FDR control is achievable under dependence with null hypothesis estimation.

Abstract

We introduce one-sided versions of Huber's contamination model, in which corrupted samples tend to take larger values than uncorrupted ones. Two intertwined problems are addressed: estimation of the mean of uncorrupted samples (minimum effect) and selection of corrupted samples (outliers). Regarding the minimum effect estimation, we derive the minimax risks and introduce adaptive estimators to the unknown number of contaminations. Interestingly, the optimal convergence rate highly differs from that in classical Huber's contamination model. Also, our analysis uncovers the effect of particular structural assumptions on the distribution of the contaminated samples. As for the problem of selecting the outliers, we formulate the problem in a multiple testing framework for which the location/scaling of the null hypotheses are unknown. We rigorously prove how estimating the null hypothesis is possible while maintaining a theoretical guarantee on the amount of the falsely selected outliers, both through false discovery rate (FDR) or post hoc bounds. As a by-product, we address a long-standing open issue on FDR control under equi-correlation, which reinforces the interest of removing dependency when making multiple testing.