The Stochastic Rayleigh diffusion model: Statistical inference and computational aspects. Applications to modelling of real cases

Abstract

In this paper, we consider a Stochastic System modelling by the Stochastic Rayleigh Diffusion Process and we discuss theoretical aspects of the latter and establish a statistical methodology to adjust it to real cases, particulary, in the field of biometry and related areas. The Rayleigh process, according to the definition of [C. Giorno, A. Nobile, L. Ricciardi, L. Sacerdote, Some remarks on the Rayleigh process, Journal of Applied Probability 23 (1986) 398–408], is examined from the perspective of the corresponding nonlinear stochastic differential equation, and from its associated probability density function we obtain the corresponding mean functions (trend function and conditional trend function), which depend of Kummer functions. The drift parameters are estimated by maximum likelihood on the basis of continuous sampling of the process and they are calculated by computational methods. We propose numerical approximations for the diffusion coefficient, from an extension of the [M. Chesney, J. Elliot, Estimating the instantaneous volatility and covariance of risky assets, Applied Stochastic Models and Data Analysis 11 (1995) 51–58] procedure to the case of nonlinear stochastic differential equations and we establish also computational procedures and simulation algorithm, that are applied to obtened simulated paths of the fitted process. The proposed methodology is applied to two studies carried out in Andalusia (Spain) on females and males life expectancy at birth, between 1944 and 2001.