Maximum likelihood estimators for generalized Cauchy processes

Abstract

Maximum likelihood estimator (MLE) for a generalized Cauchy process (GCP) is studied with the aid of the method of information geometry in statistics. Our GCP is described by the Langevin equation with multiplicative and additive noises. The exact expressions of MLEs are given for the two cases that the two types of noises are uncorrelated and mutually correlated. It is shown that the MLEs for these two GCPs are free from divergence even in the parameter region wherein the ordinary moments diverge. The MLE relations can be regarded as a generalized fluctuationdissipation theorem for the present Langevin equation. Availability of them and of some other higher order statistics is demonstrated theoretically and numerically. © 2007 American Institute of Physics.