Wavelet estimation in OFBM: Choosing scale parameter in different sampling methods and different parameter values

Abstract

Operator fractional Brownian motion (OFBM) is a multivariate extension of fractional Brownian motion and has operator self-similarity. The dependence structure across the components of OFBM is determined by the Hurst matrix H and E(XH(1)XH(1)′). In this paper, the estimators of H with wavelet method is compared in continuous sample path and discrete sample path. It is proved that with a discrete sample path, the wavelet estimator has asymptotic bias that reveals the delicate dynamic within the Hurst parameters of H, scale parameter of wavelet function, and covariance structure of XH(1). The scale parameter of wavelet function should be chosen differently to estimate Hurst parameters when Hurst index is greater than .5 than when it is less than .5 in discrete sample cases, whereas the largest scale parameter should be chosen regardless of the values of Hurst parameter when a continuous sample path is given.