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

C. Lévy-Leduc, H. Boistard, E. Moulines, M. S. Taqqu and V. A. Reisen.

Article published in Journal of Time Series Analysis, vol. 32, n. 2, p. 135-156, 2011. Download a pdf version.

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 Ma and Genton (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 than 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.