Moment-based estimation for parameters of general inverse subordinator

Abstract

In recent years the processes with anomalous diffusive dynamics have been widely discussed in the literature. The classic example of the anomalous diffusive models is the continuous time random walk (CTRW) which is a natural generalization of the random walk model. One of the fundamental properties of the classical CTRW is the fact that in the limit it tends to the Brownian motion subordinated by the so-called β-stable subordinator when the mean of waiting times is infinite. One can consider the generalization of such subordinated model by taking general inverse subordinator instead of the β-stable one as a time-change. The inverse subordinator is the first exit time of the non-decreasing Lévy process also called subordinator. In this paper we consider the Brownian motion delayed by general inverse subordinator. The main attention is paid to the estimation method of the parameters of the general inverse subordinator in the considered model. We propose a novel estimation technique based on the discretization of the subordinator’s distribution. Using this approach we demonstrate that the distribution of the constant time periods, visible in the trajectory of the considered model, can be described by the so-called modified cumulative distribution function. This paper is an extension of the authors’ previous article where a similar approach was applied, however here we focus on moment-based estimation and compare it with other popular methods of estimation. The effectiveness of the new algorithm is verified using the Monte Carlo approach.