Cellwise and Casewise Robust Multivariate Regression with Inference

TL;DR

This paper introduces a robust multivariate regression method, cellMR, that handles outliers, missing data, and high-dimensionality, along with a bootstrap inference procedure, cellBoot, validated through simulations and genomics data.

Contribution

The paper proposes the first comprehensive robust regression framework that manages both cellwise and casewise outliers, missing data, and high-dimensional settings, with valid inference.

Findings

cellMR effectively detects outliers and handles missing data in high-dimensional datasets.

cellBoot provides asymptotically valid confidence intervals robust to contamination.

Simulation and genomics studies demonstrate the method's strong finite-sample performance.

Abstract

Multivariate linear regression is a fundamental statistical task, but classical estimators such as ordinary least squares are highly sensitive to outliers. These may occur as casewise outliers that affect entire observations, or as outlying cells, that are individual contaminated entries in the predictor and/or response matrix. Moreover, modern datasets frequently contain missing values and are high-dimensional. To address these challenges we propose the cellwise multivariate regression (cellMR) estimator, a robust regression method that simultaneously accommodates casewise and cellwise outliers, missing data, and high dimensionality. The approach builds on a cellwise robust covariance estimator and uses ridge regularization for numerical stability. We further introduce cellBoot, a novel bootstrap-based inference procedure tailored to the cellMR framework. Relying on indirect inference, cellBoot provides asymptotically valid confidence intervals that are robust to casewise and cellwise contamination. We derive influence functions of the regression estimator and prove the asymptotic validity of the cellBoot confidence intervals. Simulations and a real genomics application illustrate the strong finite-sample performance of the proposed methods.