Orthogonality-based Estimation Of Nonparametric Mixed-Effect Models

In this paper, we mainly study the estimation for non-parametric mixed-effect models with longitudinal data. In the literature, there exists many effective methods for estimat-ing the fixed effects as well as the variances of the random effects and random errors in linear mixed-effect models. For example, under the assumption that the random effects and random errors both follow normal distributions, commonly used methods are maxi-mum likelihood estimation (MLE) and restricted maximum likelihood estimation (RMLE)([26]); and the BLUP estimation proposed by Robinson ([42]). However, most of those methods depend on the normality assumption, and can't give the estimation for high order moments of the random effects and random errors.[55] proposed an orthogonality-based estimation, which constructed new partial models, and then used the moment method to give the estimation for high order moments. For nonparametric mixed-effect model-s, however, most of the present methods are still based on the idea which converts the problem of spline estimation into a problem of finding the BLUP for linear mixed-effect models ([5]).In this paper, inspired by [55], we propose a new estimating method for non-parametric mixed-effect models, which is based on the orthogonal decomposition of the design matrix and spline smoothing. We use spline smoothing to give the estimation of the unknown function in non-parametric mixed-effect models and use moment estimation to estimate the high order moments. Both regression splines and penalized spline are applied to estimate the model's nonparametric part.We investigated some asymptotic properties of both the spline estimation and the moment estimation for random effects and random errors. We use the popular R lan-guage for statistical simulation, in order to confirm the effectiveness of the method and correctness of the theory.