Hurst Index Estimation for Self-Similar Processes with Long-Memory

The statistical estimation of the Hurst index is one of the fundamental problems in the literature of long-range dependent and self-similar processes. In this article, the Hurst index estimation problem is addressed for a special class of self-similar processes that exhibit long-memory, the Hermite processes. These processes generalize the fractional Brownian motion, in the sense that they share its covariance function, but are non-Gaussian. Existing estimators such as the R/S statistic, the variogram, the maximum likelihood and the wavelet-based estimators are reviewed and compared with a class of consistent estimators which are constructed based on the discrete variations of the process. Convergence theorems (asymptotic distributions) of the latter are derived using multiple Wiener-Itô integrals and Malliavin calculus techniques. Based on these results, it is shown that the latter are asymptotically more efficient than the former.