Statistica Sinica 14(2004), 763-788

POLYNOMIAL SPLINE ESTIMATION AND INFERENCE

FOR VARYING COEFFICIENT MODELS

WITH LONGITUDINAL DATA

Jianhua Z. Huang$^1$, Colin O. Wu$^2$ and Lan Zhou$^1$

$^1$University of Pennsylvania and $^2$National Heart, Lung and Blood Institute

Abstract: We consider nonparametric estimation of coefficient functions in a varying coefficient model of the form $Y_{ij}=X_i^T(t_{ij}) {\mbox{\boldmath$\beta$}}(t_{ij}) + \epsilon_i(t_{ij})$ based on longitudinal observations $\{(Y_{ij}, X_i(t_{ij}), t_{ij})$, $i=1, \ldots, n$, $j=1, \ldots, n_i\}$, where $t_{ij}$ and $n_i$ are the time of the $j$th measurement and the number of repeated measurements for the $i$th subject, and $Y_{ij}$ and $X_i(t_{ij})=(X_{i0}(t_{ij}), \ldots$, $X_{iL}(t_{ij}))^T$ for $L \geq 0$ are the $i$th subject's observed outcome and covariates at $t_{ij}$. We approximate each coefficient function by a polynomial spline and employ the least squares method to do the estimation. An asymptotic theory for the resulting estimates is established, including consistency, rate of convergence and asymptotic distribution. The asymptotic distribution results are used as a guideline to construct approximate confidence intervals and confidence bands for components of ${\mbox{\boldmath$\beta$}}(t)$. We also propose a polynomial spline estimate of the covariance structure of $\epsilon(t)$, which is used to estimate the variance of the spline estimate $\hat{\mbox{\boldmath$\beta$}}(t)$. A data example in epidemiology and a simulation study are used to demonstrate our methods.

Key words and phrases: Asymptotic normality, confidence intervals, nonparametric regression, repeated measurements, varying coefficient models.