an:02222296
Zbl 1073.62036
Huang, Jianhua Z.; Wu, Colin O.; Zhou, Lan
Polynomial spline estimation and inference for varying coefficient models with longitudinal data
EN
Stat. Sin. 14, No. 3, 763-788 (2004).
00109264
2004
j
62G08 62G20 62J10 65C40 65C05 65D07
asymptotic normality; confidence intervals; repeated measurements; varying coefficient models; CD4 depletion; HIV infection
Summary: We consider nonparametric estimation of coefficient functions in a varying coefficient model of the form \(Y_{ij}=X^T_i(t_{ij}) \beta(t_{ij})+ \varepsilon_i(t_{ij})\) based on longitudinal observations \(\{(Y_{ij},X_i (t_{ij}),t_{ij})\), \(i=1,\dots,n\), \(j=1,\dots,n_i\}\), where \(t_{in}\) 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 \(X_i(t_{ij})=(X_{i0}(t_{ij}),\dots, X_{iL} (t_{ij}))^T\) 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 \(\beta(t)\). We also propose a polynomial spline estimate of the covariance structure of \(\varepsilon(t)\), which is used to estimate the variance of the spline estimate \(\widehat\beta (t)\). A data example in epidemiologv and a simulation study are used to demonstrate our methods.