Variations and estimators for self-similarity parameters via Malliavin calculus

Abstract

Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time $1$ can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\in(1/2,2/3)$.