Nonparametric estimation of random effects densities in a linear mixed-effects model with Fourier-oscillating noise density

Abstract

This paper is devoted to the study of nonparametric estimation of random effects densities in a linear mixed-effects model. In the first case where noise distribution is fully known, we apply nonparametric deconvolution tools to construct mean consistent estimators with respect to the $L^2(\mathbb{R})$-error and then study convergence rates of the proposed estimators when noise density is Fourier-oscillating. In the second case where the random noises are assumed to be the uniform distribution on $(-a,a)$ with an unknown a > 0, we propose an estimator for a and then inherit the methodologies in the case of known noise distribution to construct necessary estimators which are also shown to be mean consistency. Some numerical results in the first case of the random noises are presented to illustrate the methodology.

Keywords:

• Random effects densities
• deconvolution
• Fourier-oscillating noise density

MSC 2010:

• 62G07
• 62G20

Acknowledgments

We would like to thank the reviewers for their kind and careful reading of the paper and for helpful comments and suggestions which led to this improved version.