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Semiparametric and nonparametric regression analysis of longitudinal data. (With comments). (English) Zbl 1015.62038
Summary: This article deals with the regression analysis of repeated measurements taken at irregular and possibly subject-specific time points. The proposed semiparametric and nonparametric models postulate that the marginal distribution for the repeatedly measured response variable \(Y\) at time \(t\) is related to the vector of possibly time-varying covariates \({\mathbf X}\) through the equations \[ E\bigl\{Y(t) \mid{\mathbf X}(t)\bigr\}= \alpha_0(t)+ \beta_0' {\mathbf X}(t) \text{ and }E\bigl\{Y(t) \mid{\mathbf X}(t)\bigr\}= \alpha_0 (t)+ \beta_0'(t) {\mathbf X}(t), \] where \(\alpha_0(t)\) is an arbitrary function of \(t,\;\beta_0\) is a vector of constant regression coefficients, and \(\beta_0(t)\) is a vector of time-varying regression coefficients. The stochastic structure of the process \(Y(\cdot)\) is completely unspecified. We develop a class of least squares type estimators for \(\beta_0\), which is proven to be \(n^{1/2}\)-consistent and asymptotically normal with simple variance estimators. Furthermore, we develop a closed-form estimator for a cumulative function of \(\beta_0(t)\), which is shown to be \(n^{1/2}\)-consistent and, on proper normalization, converges weakly to a zero-mean Gaussian process with an easily estimated covariance function.
Extensive simulation studies demonstrate that the asymptotic approximations are accurate for moderate sample sizes and that the efficiencies of the proposed semiparametric estimators are high relative to their parametric counterparts. An illustration with longitudinal CD4 cell count data taken from an HIV/AIDS clinical trial is provided.

62G08 Nonparametric regression and quantile regression
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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