Kernel Density Estimation for Random Effects Models driven by Weighted Fractional Brownian Motion

Abstract

This paper deals with the nonparametric estimation of the random effects from stochastic differential equations of the form $dX(t) = S\left( X(t), \psi\right) dt + \varepsilon d B_{a,b}(t)$, $X(0)=0$, $\eps\in (0,1]$, where $B_{a,b}$ is the weighted fractional Brownian motion. The drift function $S(\cdot,\cdot)$ is known and possibly nonlinear. Based on continuous observations of $n$ copies independent and identically distributed of $X$ (known as subjects characterized by different random effects $\psi=\psi_i$), and by using a two-stage kernel approach we formulate selected estimators of the common density $f$ of the random effects $\psi_i$. Both the $L^2$-risk bounds and the selection of the bandwidths are discussed. The proposed estimators are illustrated by numerical simulations of two concrete examples, where the random effects are either Gaussian or Beta distributed.