Asymptotic Confidence Regions for Kernel Smoothing of a Varying-Coefficient Model with Longitudinal Data

Abstract

We consider the estimation of the k + 1-dimensional nonparametric component β(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. Asymptotic distributions are established for a kernel estimate of β(t) that minimizes a local least squares criterion. These asymptotic distributions are used to construct a class of approximate pointwise and simultaneous confidence regions for β(t). Applying these methods to an epidemiological study, we show that our procedures are useful for predicting CD4 (T-helper lymphocytes) cell changes among HIV (human immunodeficiency virus)-infected persons. The finite-sample properties of our procedures are studied through Monte Carlo simulations.

Key Words:

• Asymptotic normality
• Confidence interval
• Kernel estimate
• Nonparametric regression.