Adaptive Estimation of Random-Effects Densities In Linear Mixed-Effects Model

In this paper we consider the problem of adaptive estimation of random-e ects densities in linear mixed-e ects model. The linear mixed-e ects model is de ned as Yk;j = k + ktj + "k;j where Yk;j is the observed value for individual k at time tj for k = 1; : : : ;N and j = 1; : : : ; J. Random variables ( k; k) are called random e ects and stand for the individual random variables of entity k. We denote their densities f and f and assume that they are independent of the measurement errors ("k;j ). We introduce kernel estimators of f and f and present upper risk bounds. We also compute examples of rates of convergence. The focus of this work lies on the near optimal data driven choice of the smoothing parameter using a penalization strategy in the particular case of xed interval between times tj . Risk bounds for the adaptive estimators of f and f are provided. Simulations illustrate the relevance of the methodology.