Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data

We consider the estimation of the k + 1-dimensional nonparametric component beta(t) of the varying-coefficient model Y(t) = X-T(t)beta(t) + epsilon(t) based on longitudinal observations (Y-ij, X-i(t(ij)), t(ij)), i = 1,..., n,j = i,..., 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 Rk+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 beta(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 beta(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.