Variations and Hurst index estimation for self-similar processes

We analyze the asymptotic behavior of quadratic variations for a class of non-Gaussian self-similar processes, the Hermite processes. This class is parametrized by the Hurst index and the rank/order of the process, encompassing the well-known fractional Brownian motion and the Rosenblatt process. A Hermite process of rank q with self-similarity index H has stationary, H-self-similar increments that exhibit long-memory, identical to that of the fractional Brownian motion. Furthermore, the Hermite process of order q lives in the qth Wiener chaos. Using Malliavin calculus and multiple Wiener-Itô integrals we determine the asymptotic distribution of the quadratic variations of a Hermite process of order q with self-similarity index H. Moreover, we prove a reproduction property for this class of processes in the sense that the terms appearing in the chaotic decomposition of their variations give rise to other Hermite processes of different orders and with different Hurst parameters. Furthermore, we derive the asymptotic distribution of the filtered variations of an arbitrary Hermite process, for a filter of order p. We apply our results to construct a class of strongly consistent estimators for the self-similarity parameter H from discrete observations of the process with and without longer filters. Particularly for the Rosenblatt process, we study the behavior of the asymptotic variance with respect to the order of the chosen filter for finite-difference and wavelet-based filters. Finally, we discuss classical estimators including the rescaled-range statistic, the maximum likelihood and wavelet-based estimators. We compare them with the suggested estimator in the case of the fBm and the Rosenblatt process