Wavelet Estimator of Long-Range Dependent Processes

Abstract

In this contribution, the statistical properties of the wavelet estimator of the long-range dependence parameter introduced in Abry et al. (1995) are discussed for a stationary Gaussian process. This contribution complements the heuristical discussion presented in Abry et al. (1999), by taking into account the correlation between the wavelet coefficients (which is discarded in the mentioned reference) and the bias due to the short-memory component. We derive expressions for the estimator's asymptotic bias, variance and mean-squared error as functions of the scale used in the regression and some user-defined parameters. Consistency of the estimator is obtained as long as the scale index j T goes to infinity and 2 JT /T → 0, where T denotes the sample size. Under these and some additional conditions assumed in the paper, we also establish the asymptotic normality of this estimator.