Maximum likelihood estimation of the fractal dimensions of stochastic fractals and Cramer-Rao bounds

A maximum likelihood (ML) estimator for the parameters of Gaussian versions of the fractionally differenced white noise process is developed. A closed-form expression for the likelihood equation is given, and Cramer-Rao bounds are computed for finite size sample data sets. It is shown how the theory can be extended to the case where the fractionally differenced white noise process is observed in the presence of white noise. The results obtained with this ML approach are satisfactory, with a mean square error which is very close to the theoretically computed Cramer-Rao bound