Adaptive estimation of marginal random-effects densities in linear mixed-effects models

Abstract

In this paper we consider the problem of adaptive estimation of random-effects densities in linear mixed-effects model. The linear mixed-effects model is defined as Y k,j = α k + β k t j + ɛ k,j , where Y k,j is the observed value for individual k at time t j for k = 1, …,N and j = 1, …, J. Random variables (α k , β k ) are called random effects and stand for the individual random variables of entity k. We denote their densities f α and f β and assume that they are independent of the measurement errors (ɛ k,j ). We introduce kernel estimators of f α and f β 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 fixed interval between times t j . Risk bounds for the adaptive estimators of f α and f β are provided. Simulations illustrate the relevance of the methodology.