Robust estimation of the scale and of the autocovariance function of Gaussian short and long-range dependent processes

TL;DR

This paper analyzes the asymptotic properties of robust autocovariance estimators for Gaussian processes, demonstrating their advantages over classical estimators especially in the presence of outliers and for long-range dependence.

Contribution

It establishes the asymptotic normality of robust autocovariance estimators for Gaussian processes under short and long-range dependence, with explicit variance expressions and practical validation.

Findings

Robust estimators are asymptotically normal at rate √n in short-range dependence.

In long-range dependence, the limit distribution is Gaussian for H < 3/4 and non-Gaussian for H > 3/4.

Monte Carlo simulations and real data analysis support the robustness and effectiveness of the estimators.

Abstract

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well-known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time-series modeling. In this paper, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Genton and Ma (2000) are established for Gaussian processes, with either short-range or long-range dependence. It is shown in the short-range dependence setting that this robust estimator is asymptotically normal at the rate $\sqrt{n}$, where $n$ is the number of observations. An explicit expression of the asymptotic variance is also given and compared to the asymptotic variance of the classical autocovariance estimator. In the long-range dependence setting, the limiting distribution displays the same behavior than that of the classical autocovariance estimator, with a Gaussian limit and rate $\sqrt{n}$ when the Hurst parameter $H$ is less 3/4 and with a non-Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when $H \in (3/4,1)$. Some Monte-Carlo experiments are presented to illustrate our claims and the Nile River data is analyzed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.