# 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 $Y_{k,j} = \alpha_k + \beta_k t_j + \varepsilon_{k,j}$ where $Y_{k,j}$ is the observed value for individual $k$ at time $t_j$ for $k=1,\ldots, N$ and $j=1,\ldots , J$. Random variables $(\alpha_k, \beta_k)$ are known as random effects and stand for the individual random variables of entity $k$. We denote their densities $f_\alpha$ and $f_\beta$ and assume that they are independent of the measurement errors $(\varepsilon_{k,j})$. We introduce kernel estimators and present upper risk bounds. We also give rates of convergence. The focus of this work lies on the optimal data driven choice of the smoothing parameter using a penalization strategy in the particular case of fixed interval between times $t_j$.
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https://hal.archives-ouvertes.fr/hal-00958905
Contributor : Gwennaëlle Mabon <>
Submitted on : Friday, April 17, 2015 - 12:02:24 PM
Last modification on : Thursday, April 11, 2019 - 4:02:09 PM
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Gwennaëlle Mabon. Adaptive estimation of marginal random-effects densities in linear mixed-effects models. Mathematical Methods of Statistics, Allerton Press, Springer (link), 2015, 24 (2), pp.81-102. ⟨10.3103/S1066530715020015⟩. ⟨hal-00958905v4⟩

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