The Property Of Cross-validated Kernel Estimates Of Varying-Coefficient Regression Model With Longitudinal Data

This paper considers the estimation of the k+1-dimensional nonparametric componentβ(t) of the varying-coefficient model Y(t) = X^T(t)β(t) +ε(t) based on longitudinal observations Y_ij,X_i(t_ij),t_ij), i = 1,…,n, j = 1,…,n_i, where t_ij is the jth observed design time point t of the ith subject and Y_ij and X_i(t_ij) are the real-valued outcome and R^k+1 valued covariate vectors of the ith subject at t_ij. The subjects are independently selected, but the repeated measurements within subject are possibly correlated.One very important method in estimating the coefficient functions in varying-coefficient model is the kernel estimator. In this approach, the choice of bandwidth is crucial. A leave-one-subject-out ordinary cross-validation criterion is used for selecting data-driven bandwidth. Although the Cross-Validation criterion was used to select the bandwidth in the kernel estimate of time-varying coefficient models with longitu-dinal data in the literature, within author's knowledge, there was nobody to investigate the theoretical properties of the complicated estimates. Under some mild assumptions, author investigates this issue and derives the rate of convergence of the cross-validated smoothing parameters to their optimal benchmark values, and establishes the asymptotic normality of the resulting nonparametric estimator. These asymptotic results give useful insights into the reliability of the A leave-one-subject-out ordinary cross-validation criterion. The finite sample performance of the cross-validation smoothing parameter selector are investigated through a simulation study.