A regularised estimator for long-range dependent processes

There is significant interest in long-range dependent processes since they occur in a wide range of phenomena across different areas of study. Based on the available models capable of describing long-range dependence, various parameter estimation methods have been developed. In this paper we revisit the maximum likelihood estimator and its computationally efficient approximations: Whittle’s Estimator and the Circulant Embedding Estimator. In particular, this paper proves the asymptotic properties of the Circulant Embedding estimator and establishes the asymptotic equivalence between the three estimators. Furthermore, it is shown that the three methods are ill-conditioned and thus highly sensitive to the presence of measurement errors. Finally, we introduce a regularisation method that improves the performance of the maximum likelihood methods when the datasets have been largely contaminated with errors.