ABSTRACT

Variance function estimation and semiparametric regression are important problems in many contexts with a wide range of applications.  In this talk I will present some new results on these two problems. A consistent theme is the use of a difference based method. I will begin with a minimax analysis of the variance function estimation in heteroscedastic nonparametric regression. The results indicate that, contrary to the common practice, it is often not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. The results also correct the optimal rate claimed in Hall and Carroll (1989, JRSSB). I will then consider adaptive estimation of the variance function using a wavelet thresholding approach. A data-driven estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. The variance function estimator is shown to be nearly optimally adaptive to the smoothness of both the mean and variance functions. Finally I will discuss a difference based procedure for semiparametric partial linear models. The estimation procedure is optimal in the sense that the estimator of the linear component is asymptotically efficient and the estimator of the nonparametric component is minimax rate optimal. Some numerical results will also be discussed.