Wavelet-based Estimator for the Hurst Parameters of Fractional Brownian Sheet

Abstract

It is proposed a class of statistical estimators $\hat{H}=({\hat{H}}_1,\dots ,{\hat{H}}_d)$ for the Hurst parameters $H=(H_1,\dots ,H_d)$ of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging (DTI) of nuclear magnetic resonance (NMR). Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when $H_i\ge 1/2$, the estimators are efficient, and when $H_i<1/2$, there are some bias.