Robust estimation for high-dimensional time series with heavy tails

TL;DR

This paper introduces a robust LAD regression method for high-dimensional, heavy-tailed time series data, achieving improved risk bounds and demonstrating superior performance over classical LAD in simulations and real data applications.

Contribution

It proposes a Catoni-type truncated minimization framework for LAD regression in heavy-tailed, dependent data, with theoretical risk bounds and practical validation.

Findings

The new estimator reduces excess risk in heavy-tailed time series.

Simulations show classical LAD risk can blow up, while the new method remains stable.

Application to real data demonstrates better fit than classical LAD.

Abstract

We study in this paper the problem of least absolute deviation (LAD) regression for high-dimensional heavy-tailed time series which have finite $\alpha$-th moment with $\alpha \in (1,2]$. To handle the heavy-tailed dependent data, we propose a Catoni type truncated minimization problem framework and obtain an $\mathcal{O}\big( \big( (d_1+d_2) (d_1\land d_2) \log^2 n / n \big)^{(\alpha - 1)/\alpha} \big)$ order excess risk, where $d_1$ and $d_2$ are the dimensionality and $n$ is the number of samples. We apply our result to study the LAD regression on high-dimensional heavy-tailed vector autoregressive (VAR) process. Simulations for the VAR($p$) model show that our new estimator with truncation are essential because the risk of the classical LAD has a tendency to blow up. We further apply our estimation to the real data and find that ours fits the data better than the classical LAD.