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:

62M09 | Non-Markovian processes: estimation |

62F12 | Asymptotic properties of parametric estimators |

60G18 | Self-similar processes |

60H07 | Stochastic calculus of variations and the Malliavin calculus |