Adaptive Huber Regression on Markov-dependent Data

TL;DR

This paper extends Adaptive Huber Regression to Markov-dependent data, showing how dependence affects the choice of robustification parameters and regression estimation in high-dimensional time-series.

Contribution

It provides theoretical justification for AHR under Markov dependence, incorporating the spectral gap of the chain into the analysis.

Findings

Markov dependence influences the robustification parameter adaptation.

Sample size adjustment depends on the spectral gap of the Markov chain.

The method effectively handles heavy-tailed errors in dependent data.

Abstract

High-dimensional linear regression has been intensively studied in the community of statistics in the last two decades. For the convenience of theoretical analyses, classical methods usually assume independent observations and sub-Gaussian-tailed errors. However, neither of them hold in many real high-dimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer. Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the issue of heavy-tailed errors. They discover that the robustification parameter of the Huber loss should adapt to the sample size, the dimensionality, and the moments of the heavy-tailed errors. We progress in a vertical direction and justify AHR on dependent observations. Specifically, we consider an important dependence structure -- Markov dependence. Our results show that the Markov dependence impacts on the adaption of the robustification parameter and the estimation of regression coefficients in the way that the sample size should be discounted by a factor depending on the spectral gap of the underlying Markov chain.