MAXIMUM-LIKELIHOOD-ESTIMATION OF THE PARAMETERS OF DISCRETE FRACTIONALLY DIFFERENCED GAUSSIAN-NOISE PROCESS

A maximum likelihood estimation procedure is constructed for estimatin g the parameters of discrete fractionally differenced Gaussian noise f rom an observation set of finite size N. The procedure does not involv e the computation of any matrix inverse or determinant. It requires N2 /2 + O(N) operations. The expected value of the loglikelihood function for estimating the parameter d of fractionally differenced Gaussian n oise (which corresponds to a parameter of the equivalent continuous-ti me fractional Brownian motion related to its fractal dimension) is sho wn to have a unique maximum with the range of allowable values of d. T he maximum occurs at the true value of d. A Cramer-Rao bound on the va riance of any unbiased estimate of d obtained from a finite size obser vation set is derived. It is shown experimentally that the maximum lik elihood estimate of d is unbiased and efficient when finite size data sets are used in the estimation procedure. The proposed procedure is a lso extended to deal with noisy observations of discrete fractionally differenced Gaussian noise.