Nonparametric estimation of random-effects densities in linear mixed-effects model

Abstract

Abstract. We consider a linear mixed-effects model where Y k,j = α k +β k tj +ε k,j is the observed value for individual k at time tj, k = 1, . . . , N , j = 1, . . . , J. The random effects α k , β k are independent identically distributed random variables with unknown densities fα and f β and are independent of the noise. We develop nonparametric estimators of these two densities, which involve a cutoff parameter. We study their mean integrated square risk and propose cutoff-selection strategies, depending on the noise distribution assumptions. Lastly, in the particular case of fixed interval between times tj, we show that a completely data driven strategy can be implemented without any knowledge on the noise density. Intensive simulation experiments illustrate the method.

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

“…Compared to Comte and Samson (2012), this inequality differs from the last term which is smaller than theirs, for s N ≡ 1. The first term of variance with order m/N is the bound we would have if we were in a direct density estimation context.…”

“…In Table 4, we have reported the results of the simulations with Comte and Samson (2012) procedure computed over the same samples used in Tables 2 and 3. We have chosen to compare standard Gaussian distribution and Gamma distribution as examples.…”

“…Moreover the times t j are chosen as t j = j∆ with ∆ = 2 and J = 6 as in Comte and Samson (2012). 5,26].…”

“…Model (1) has been considered by Comte and Samson (2012). Let us emphasize the novelty of our paper.…”

“…Let us emphasize the novelty of our paper. Comte and Samson (2012) derive theoretical properties of the nonparametric estimators of f α and f β only when the error distribution is known. In that particular case they establish oracle inequalities which ensure that their method is adaptive.…”