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Hurst index estimation for self-similar processes with long-memory. (English) Zbl 1196.62112
Duan, Jinqiao (ed.) et al., Recent development in stochastic dynamics and stochastic analysis. Dedicated to Zhi-Yuan Zhang on the occasion of his 75th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-4277-25-9/hbk). Interdisciplinary Mathematical Sciences 8, 91-117 (2010).
Summary: 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 discrete variations of the process. Convergence theorems (asymptotic distributions) of the latter are derived using multiple Wiener-ItĂ´ integrals and Malliavin calculus techniques. Based on these results, it is shown that the latter are asymptotically more efficient than the former. For the entire collection see [Zbl 1191.60005].

MSC:
62M09Non-Markovian processes: estimation
62F12Asymptotic properties of parametric estimators
60G18Self-similar processes
60H07Stochastic calculus of variations and the Malliavin calculus