On MLE methods for dynamical systems with fractionally differenced noise spectra

Maximum likelihood is an attractive estimator for linear systems with finite order. In the case of fractionally differenced processes, the maximum likelihood estimator becomes numerically intractable for large data sets. An algorithm for the estimation of the fractal dimension of a process that addresses the ill-conditioning of its covariance matrix is proposed. The algorithm reduces the variance of the fractal dimension estimate by segmenting the data into several sequences of relatively small length. The algorithm possesses better numerical properties than the ones proposed in the literature. An extension to the algorithm is proposed in order to cover ARFIMA models and its convergence properties are discussed. While no guarantee of its convergence is offered, the algorithm's good behaviour is shown in simulations.