Single-index models, in the simplest form E(y|x)=g(xTb), generalize linear models by allowing flexible curvatures captured by the unknown function g(.), and at the same time, retain the same easy interpretability as in linear models, given the index parameter vector b that form the linear index xTb. In addition, compared with fully nonparametric models, single-index models avoid the “curse of dimensionality”. This dissertation consists of two essays on single-index models.
The first essay is concerned with estimation of single-index varying coefficient models. Varying coefficient models assume that the regression coefficients vary with some threshold variables. Previous research focused on the case of a single threshold variable. It is common for the coefficients to depend on multiple threshold variables but the resulting model is difficult to estimate. Single-index coefficient models alleviate the difficulty by modeling each coefficient by a function of an index. Existing estimation approaches employ kernel smoothing or local linear approximation of the coefficient functions (Xia and Li, 1999; Cai, Fan and Yao, 2003) which entail heavy computational burden. Also, implementation of different bandwidths for different coefficient functions to allow different smoothness is difficult for local approaches. We propose a penalized spline approach to estimating single-index coefficient models that not only allows different smoothness for different coefficient functions but also is computationally fast. Asymptotic theory is established under dependency. Numerical studies demonstrate the proposed approach.
The second essay is on single-index quantile regression. Nonparametric quantile regression with multivariate covariates is often a difficult estimation problem due to the “curse of dimensionality”. Single-index quantile regression, where the conditional quantile is modeled by a nonparametric link function of a linear combination of covariates, can reduce the dimensionality of the estimation problem while retaining both the flexibility of a nonparametric model and easy interpretability of a simple linear model. We extend the local linear approach of Yu and Jones (1998) to estimation of single-index quantile models and introduce an iterative algorithm. Large sample properties of estimators for both the nonparametric part and the parametric part are studied. Simulation results together with real data applications show promise of the new approach.