Estimating minimum effect with outlier selection

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.

Confidence regions and minimax rates in outlier-robust estimation on the probability simplex

“…Computational intractability of Tukey's median motivated a large number of studies that aimed at designing computationally tractable methods with nearly optimal statistical guarantees. Many of these works went beyond Huber's contamination by considering parameter contamination models (Bhatia et al, 2017;Collier and Dalalyan, 2017;Carpentier et al, 2018), oblivious contamination (Feng et al, 2014;Lai et al, 2016) or adversarial contamination (Diakonikolas et al, 2016;Balakrishnan et al, 2017;Diakonikolas et al, 2017Diakonikolas et al, , 2018. Interestingly, in the problem of estimating the Gaussian mean, it was proven that the minimax rates under adversarial contamination are within a factor at most logarithmic in n and k of the minimax rates under Huber's contamination 6 .…”