inference for fractional stochastic processes with random effects: parametric and non parametric approach

Abstract

Stochastic di erential equation models with random e ects are increasingly used in the biomedical elds and have proved to be adequate tools for the study of repeated measurements collected on series of subjects. These models allow the quanti cation of both between and within subject variation. Performing parametric inference for such models, using discrete (or continuous) time data, is a challenging problem for two reasons: First, the state likelihood is a product of transition densities which are rarely known. Second, the marginalization required to construct this likelihood is an (often multidimensional) integral, which rarely has a closed-form solution. We provide a class of estimators for Stochastic di erential equations (SDE's) with random e ects and examine their asymptotic behaviour. We are concerned with SDE's with nonlinear drift and generalized random e ects, for which a simulation study is given to highlight the performance of the proposed estimators. We extend the existing results of statistical inference for random e ects models to include the SDE's with random e ects driven by fractional Brownian motion (fBm). The incorporation of the fBm within our models is of great interest, since it accounts for dependency of increments of the noisy term. This is the case of long-memory phenomena arising in variety of di erent scienti c elds, including hydrology, biology, medicine, economics and tra c network. We consider linear fractional stochastic di erential equations with random e ects, provide estimators of the common density of random e ects, and examine their asymptotic properties. Two types of estimators are considered: kernel density estimators and histogram estimators. Most of our results are illustrated by relevant examples.