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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.

MSC:

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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