Mathematical Methods of Statistics

, Volume 24, Issue 2, pp 81–109 | Cite as

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

Article

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 Yk,j = αk + βktj + ɛk,j, where Yk,j is the observed value for individual k at time tj for k = 1, …,N and j = 1, …, J. Random variables (αk, βk) are called random effects and stand for the individual random variables of entity k. We denote their densities fα and fβ and assume that they are independent of the measurement errors (ɛk,j). We introduce kernel estimators of fα and fβ 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 tj. Risk bounds for the adaptive estimators of fα and fβ are provided. Simulations illustrate the relevance of the methodology.

Keywords

adaptive estimation nonparametric density estimation deconvolution linear mixed-effects model random effect density mean squared risk 

2000 Mathematics Subject Classification

primary 62G07 

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Copyright information

© Allerton Press, Inc. 2015

Authors and Affiliations

  1. 1.CREST-ENSAE, Malakoff; MAP5Univ. Paris DescartesParisFrance

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