Central limit theorem for the robust log-regression wavelet estimation of the memory parameter in the Gaussian semi-parametric context

Abstract

In this paper, we study robust estimators of the memory parameter $d$ of a (possibly) non-stationary Gaussian time series with generalized spectral density $f$. This generalized spectral density is characterized by the memory parameter $d$ and by a function $f^{\ast}$ which specifies the short-range dependence structure of the process. Our setting is semi-parametric since both $f^{\ast}$ and $d$ are unknown, and $d$ is the only parameter of interest. The memory parameter $d$ is estimated by regressing the logarithm of the estimated variance of the wavelet coefficients at different scales. The two estimators of $d$ that we consider are based on robust estimators of the variance of the wavelet coefficients, namely the square of the scale estimator proposed by Rousseeuw and Croux [J. Amer. Statist. Assoc. 88 (1993) 1273–1283] and the median of the square of the wavelet coefficients. We establish a central limit theorem, for these robust estimators as well as for the estimator of $d$, based on the classical estimator of the variance proposed by Moulines, Roueff and Taqqu [Fractals 15 (2007) 301–313]. Some Monte-Carlo experiments are presented to illustrate our claims and compare the performance of the different estimators. The properties of the three estimators are also compared to the Nile river data and the Internet traffic packet counts data. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the memory parameter $d$ of Gaussian time series.