Research On Semi-varying Coefficient Models And Single-index Models

Semiparametric regression models can not only retain the great adaptability ofnonparametric regression models, but also avoid “curse of dimensionality”, and thushave been extensively studied and been applied in various fields in recent years. In thisthesis, our studies mainly focus on two of semiparametric models, semivaryingcoefficient model and single-index model.For semivarying coefficient model with longiudinal data, we consider the empiricallikelihood inference for the parametric coefficients in this model. Combining the profileleast squares technique and empirical likelihood, we construct an empirical likelihoodratio statistic for the parametric coefficient and we prove the statistic to beasymptotically standard chi-squared distributed. Thus the nonparametric Wilksphenonmenon is derived. The confidence region for the parametric coefficient based onthe empirical likelihood method is obtained. Moreover, we define the estimator of theparametric coefficient by maximizing the empirical likelihood ratio function. Theasymptotic property of the estimator is also considered. A simulation study is conductedto demonstrate the numerical property of the proposed empirical likelihood ratiostatistics and the estimators of the parametric coefficients.For single-index model with covariates missing at random, to deal with the constraintsatisfied by the index parameter, we employ the “delete-one-component” method totransform the parameter and introduce an inverse probability weighted (IPW) estimatingequation to do the model estimation. The asymptotic properties of the IPW estimatorsare considered when the selection probability is known or unknown. For the indexparameters in the single-index model, it is shown that the IPW estimator with estimatedselection probability has a smaller asymptotic variance than the estimator with the trueselection probablity, and thus the former one is more efficient. Therefore, the estimatorsof the index parameters in single-index models with missing covariates have theimportant Horvitz-Thompson property. However, the IPW estimators for the linkfunction have the same asymptotic distribution whether the missing probability isknown or not. That is, for the link function in single-index models with missingcovariates, the Horvitz-Thompson property doesn’t hold any more. The numericalproperties of the proposed estimators are demonstrated by some simulations.In Chapter4, we reconsider the asymptotic property of the estimators of the index parameters for single-index models based on the estimating equation introduced byChang et al.(2010), derive that the estimator has the consistent asymptotic variancewith the standard results in the literature, and correct an error in Chang et al.(2010).Moreover, to avoid the “undersmoothing” involved in the estimating equation, wepropose a bias-corrected centered estimating equation and prove that the estimator ofthe index parameters still has root-n consistency when the bandwidth is optimal. Theasymptotic normality of the estimator is also derived. Some numerical simulations and areal data example are used to discuss the numerical properties of the estimators andverify our theoretical results.Moreover, we further consider the variable selection of the index parameter forsingle-index models. By using the adaptive lasso method, we propose a penalizedcentered estimating equation to select and estimate the index parameter simultaneously.And this method can not only select the true model consistently, but also estimate thenonzero components as well as when the true model is known. Thus the estimatorsbased on the penalized centered estimating equation have the oracle property. Asimulation study is conducted to consider the numerical property of the estimators andthe selection of the tuning parameters.