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

Gwennaëlle Mabon (1, 2)

1 CREST - Centre de Recherche en Économie et Statistique
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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 $t_j$.

Mots clés: Adaptive estimation. Nonparametric density estimation. Deconvolution. Linear mixed-effects model. Random effect density. Mean square risk

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Gwennaëlle Mabon. Adaptive estimation of random-effects densities in linear mixed-effects model. 2014. ⟨hal-00958905v2⟩