A Difference Based Method in Nonparametric Function Estimation 
Lie Wang , Ph.D. Candidate
The Wharton School, University of Pennsylvannia
3:00 p.m.; February 6, 2008
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 datadriven
estimator is constructed by applying wavelet thresholding to the squared
firstorder 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.
