  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* which specifies the short-range dependence
structure of the process. Our setting is semi-parametric since both f* 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 (1993) 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 (2007). 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 on 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.
