Modified cumulative distribution function in application to waiting time analysis in CTRW scenario

Abstract

The continuous time random walk model plays an important role in modeling of so called anomalous diffusion behaviour. One of the specific property of such model are constant time periods visible in trajectory. In the continuous time random walk approach they are realizations of the sequence called waiting times. The main attention of the paper is paid on the analysis of waiting times distribution. We introduce here novel methods of estimation and statistical investigation of such distribution. The methods are based on the modified cumulative distribution function. In this paper we consider three special cases of waiting time distributions, namely $\alpha$-stable, tempered stable and gamma. However the proposed methodology can be applied to broad set of distributions - in general it may serve as a method of fitting any distribution function if the observations are rounded. The new statistical techniques we apply to the simulated data as well as to the real data describing $CO_2$ concentration in indoor air.