Robust estimation of the memory parameter of Gaussian time series using wavelets

We propose in this paper 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. The memory parameter d is estimated by regressing the logarithm of the estimated variance of the wavelet coefficients at different scales. The two robust 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 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. The properties of these estimators are also compared on publicly available Internet traffic packet counts data.