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Kernel-based generalized cross-validation in non-parametric mixed-effect models. (English) Zbl 1194.62058

A nonparametric additive mixed-effect model is considered of the form \(y_i=g(t_i)+z_i^T b +\varepsilon_i\), where \(g\) is an unknown smooth function, \(t_i\) and \(z_i\) are covariates, \(b\) is the \(N(0,\sigma^2\Sigma^{-1})\) vector of random effects, and \(\varepsilon_i\) are \(N(0,\sigma)\) errors independent of \(b\). Here \(\Sigma=\Sigma(\varphi)\), where \(\varphi\) is a tuning parameter. A kernel-based weighted least squares algorithm is considered for the estimation of \(g\). A generalized cross-validation (GCV) algorithm is proposed for the choice of the bandwidths \(h\) and \(\varphi\). The asymptotic optimality of this choice is demonstrated. Repeated measures, clustered observations and a mixture of these two models are considered as examples. Results of simulations and an application to pigs weights data are presented.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)

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