Asymptotic Normality of a Hurst Parameter Estimator Based on the Modified Allan Variance

In order to estimate the memory parameter of Internet traffic data, it has been recently proposed a log-regression estimator based on the so-called modified Allan variance (MAVAR). Simulations have shown that this estimator achieves higher accuracy and better confidence when compared with other methods. In this paper we present a rigorous study of the MAVAR log-regression estimator. In particular, under the assumption that the signal process is a fractional Brownian motion, we prove that it is consistent and asymptotically normally distributed. Finally, we discuss its connection with the wavelets estimators. 1. Introduction It is well known that different kinds of real data (hydrology, telecommunication networks, economics, and biology) display self-similarity and long-range dependence (LRD) on various time scales. By self-similarity we refer to the property that a dilated portion of a realization has the same statistical characterization as the original realization. This can be well represented by a self-similar random process with a given scaling exponent (Hurst parameter). The long-range dependence, also called long memory, emphasizes the long-range time correlation between past and future observations and it is thus commonly equated to an asymptotic power law behaviour of the spectral density at low frequencies or, equivalently, to an asymptotic power-law decrease of the autocovariance function, of a given stationary random process. In this situation, the memory parameter of the process is given by the exponent characterizing the power law of the spectral density. (For a review of historical and statistical aspects of the self-similarity and the long memory see [1].) Though a self-similar process cannot be stationary (and thus nor LRD), these two properties are often related in the following sense. Under the hypothesis that a self-similar process has stationary (or weakly stationary) increments, the scaling parameter enters in the description of the spectral density and covariance function of the increments, providing an asymptotic power law with exponent . Under this assumption, we can say that the self-similarity of the process reflects on the long-range dependence of its increments. The most paradigmatic example of this connection is provided by the fractional Brownian motion and by its increment process, the fractional Gaussian noise [2]. In this paper we will consider the problem of estimating the Hurst parameter of a self-similar process with weakly stationary increments. Among the different techniques introduced in the literature, we will