## Local polynomial mixed-effects models for longitudinal data.(English)Zbl 1048.62048

Summary: We consider a nonparametric mixed-effects model $$y_i(t_{ij})= (t_{ij})+v_i(t_{ij})+\varepsilon_i(t_{ij})$$, $$j=1,2,\dots,n_i$$; $$i=1,2,\dots,n$$, for longitudinal data. We propose combining local polynomial kernel regression and linear mixed-effects (LME) model techniques to estimate both fixed effects (population) curves $$\eta(t)$$ and random-effects curves $$v_i(t)$$. The resulting estimator, called the local polynomial LME (LLME) estimator, takes the local correlation structure of the longitudinal data naturally into account. We also propose new bandwidth selection strategies for estimating $$\eta(t)$$ and $$v_i(t)$$.
Simulation studies show that our estimator for $$\eta(t)$$ is superior to the existing estimators in the sense of mean squared errors. The asymptotic bias, variance, mean squared errors, and asymptotic normality are established for the LLME estimators of $$\eta(t)$$. When $$n_i$$ is bounded and $$n$$ tends to infinity, our LLME estimator converges in a standard nonparametric rate, and the asymptotic bias and variance are essentially the same as those of the kernel generalized estimating equation estimator proposed by X. Lin and R. J. Carroll [ibid. 95, No. 450, 520–534 (2000; Zbl 0995.62043)]. But when both $$n_i$$ and $$n$$ tend to infinity, the LLME estimator is consistent with a slower rate of $$n^{1/2}$$ compared to the standard nonparametric rate, due to the existence of within-subject correlations of longitudinal data. We illustrate our methods with an application to a longitudinal dataset.

### MSC:

 62G08 Nonparametric regression and quantile regression 62P10 Applications of statistics to biology and medical sciences; meta analysis 62G20 Asymptotic properties of nonparametric inference

Zbl 0995.62043
Full Text: