Asymptotic self-similarity and wavelet estimation for long-range dependent fractional autoregressive integrated moving average time series with stable innovations

Methods for parameter estimation in the presence of long-range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long-range dependence in the case of heavy-tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/&agr; for long-range dependent FARIMA time series with symmetric &agr;-stable (1 < &agr; < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self-similar with parameter H. When the parameter &agr; is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H - 1/&agr;. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet-based estimation method used here is semi-parametric, it allows for a more robust treatment of long-range dependent data than parametric methods. Copyright 2005 Blackwell Publishing Ltd.