Semiparametric Mean Covariance Analysis Of Time Varging-coefficient Single-index Models For Longitudinal Data

Longitudinal data arise frequently in agriculture, industry, biomedicine, epidemi-ology, social economy and so on. Hence, the research on longitudinal data has been the focus of attention of domestic and foreign statistical scholars. An important fea-ture of longitudinal data is within-subject correlation, if we ignore this correlation, it may lead to the unefficient estimation of regression coefficients. Therefore, estimating the covariance structure is also an essential part of longitudinal data analysis. Varying-coefficient single-index models (VCSIM) provide a good balance between parametric and nonparametric models. Concretely, they overcome the disadvantage of "the curse of dimensionality" of nonparametric models, at the same time, reserve the advantages of parsimony and intuitive interpretations of parametric models. Hence semiparamet-ric mean-covariance analysis of VCSIMs for longitudinal data is considered in this paper.First of all, various statistical models and the research status of covariance struc-ture of longitudinal data are introduced. Secondly, VCSIMs for longitudinal data and their estimation methods are introduced. Concretely, based on the variance-correlation decomposition, a semiparametric model is established for the covariance structure.The variance function and correlation structure parameters are estimated by the gener-alized estimating equation (GEE) technique and the quasi-likelihood approach respec-tively. Then based on the estimated covariance structure, the estimates of single-index parameters and unknown functions in the mean are obtained by a profile iteration ap-proach. Lastly, under some assumptions, the large sample properties of the estimators are established. The validity of the proposed estimation procedure is also verified by simulations and real data analysis.