This article deals with statistical inferences based on the varying-coefficient models proposed by Hastie and Tibshirani. Local polynomial regression techniques are used to estimate coefficient functions, and the asymptotic normality of the resulting estimators is established. The standard error formulas for estimated coefficients are derived and are empirically tested. A goodness-of-fit test technique, based on a nonparametric maximum likelihood ratio type of test, is also proposed to detect whether certain coefficient functions in a varying-coefficient model are constant or whether any covariates are statistically significant in the model. The null distribution of the test is estimated by a conditional bootstrap method. Our estimation techniques involve solving hundreds of local likelihood equations. To reduce the computational burden, a one-step Newton-Raphson estimator is proposed and implemented. The resulting one

A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in Fan and Li (2001a). It has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance. Such a property is called an oracle property. The proposed procedures were illustrated in the context of linear regression, robust linear regression and generalized linear models. In this paper, the nonconcave penalized likelihood approach is extended further to the Cox proportional hazards model and the Cox proportional hazards frailty model, two commonly used semi-parametric models in survival analysis. As a result, new variable selection procedures for these two commonly-used models are proposed. It is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function. Further, with a proper choice of

High dimensionality comparable to sample size is common in many statistical problems. We examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases. Motivated by the Arbitrage Pricing Theory in finance, a multi-factor model is employed to reduce dimensionality and to estimate the covariance matrix. The factors are observable and the number of factors K is allowed to grow with p. We investigate the impact of p and K on the performance of the model-based covariance matrix estimator. Under mild assumptions, we have established convergence rates and asymptotic normality of the model-based estimator. Its performance is compared with that of the sample covariance matrix. We identify situations under which the factor approach increases performance substantially or marginally. The impacts of covariance matrix estimation on optimal

Varying-coefficient partially linear models are frequently used in statistical modelling, but their estimation and inference have not been systematically studied. This paper proposes a profile least-squares technique for estimating the parametric component and studies the asymptotic normality of the profile least-squares estimator. The main focus is the examination of whether the generalized likelihood technique developed by Fan et al. is applicable to the testing problem for the parametric component of semiparametric models. We introduce the profile likelihood ratio test and demonstrate that it follows an asymptotically 2 distribution under the null hypothesis. This not only unveils a new Wilks type of phenomenon, but also provides a simple and useful method for semiparametric inferences. In addition, the Wald statistic for semiparametric models is introduced and demonstrated to possess a sampling property similar

When estimating a mean regression function and its derivatives, locally weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimating curves with a complicated structure, a variable bandwidth is desirable. Furthermore, the bandwidth should be indicated by the data themselves. Recent developments in nonparametric smoothing techniques inspired us to propose such a datadriven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. The idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator. The procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives. The resulting estimation