Maximum likelihood estimators for generalized Cauchy processes

Abstract

Maximum likelihood estimator ͑MLE͒ for a generalized Cauchy process ͑GCP͒ 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.

Publisher's Note: Stationary and dynamical properties of information entropies in nonextensive systems [Phys. Rev. E77, 031133 (2008)]

“…(2). Recently Konno and Watanabe [17] have made a detailed study of the Fisher information g ij in the stationary state of the Langevin model subjected to cross-correlated additive and multiplicative noise, related discussion being given in Sec. 4.…”

Superstatistics and System Identification for a Class of Generalized Cauchy Processes

“…The second order models III, IV and V in section 4 can be also identified completely by the procedure of inferring them only from time series data with the aid Proceedings of the 45th ISCIE International Symposium on Stochastic Systems Theory and Its Applications Okinawa, Nov. [1][2]2013 of the information on the correlation functions associated with them. On the other hand, the feasibility to identify models of third order nonlinear systems only from the time series data will be discussed in section 5.…”

“…This is because the ordinary moment diverges for small values of b < 1/2. Also, the MLE [1] is free from the mathematical divergence, it contains the non-canonical parameter a.…”

“…This gives a slow stochastic process with the Cauchy distribution in Eq. (1). However, there are many other stochastic processes which give a class of the Cauchy type distributions.…”

“…In our previous paper [1], we have studied a generalized Cauchy process which gives the probability density function in the form…”