Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. (English) Zbl 1064.62523

Summary: We consider the estimation of the \((k+1)\)-dimensional nonparametric component \(\beta(t)\) of the varying-coefficient model \(Y(t)=\mathbf X^T(t)\beta(t)+\epsilon(t)\) based on longitudinal observations \((Y_{ij},\mathbf X_i(t_{ij}),t_{ij}),\;i=1,\cdots,n, j=1,\cdots,n_i\), where \(t_{ij}\) is the \(j\)th observed design time point \(t\) of the \(i\)th subject and \(Y_{ij}\) and \(\mathbf X_i(t_{ij})\) are the real-valued outcome and \(\mathbb R^{k+1}\) valued covariate vectors of the \(i\)th 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.


62G08 Nonparametric regression and quantile regression
62E20 Asymptotic distribution theory in statistics
62G07 Density estimation
62G15 Nonparametric tolerance and confidence regions
62P10 Applications of statistics to biology and medical sciences; meta analysis
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