Efficient semiparametric estimator for heteroscedastic partially linear models
1 Department of Statistics, Texas A&M University, College Station, Texas 77843, U.S.A. ma{at}stat.tamu.edu, 2 Institute of Statistical Science, Academia Sinica, 128 Academia Road, Taipei 115, Taiwan jmchiou{at}stat.sinica.edu.tw, 3 Department of Statistics, Texas A&M University, College Station, Texas 77843, U.S.A. nwang{at}stat.tamu.edu
We study the heteroscedastic partially linear model with an unspecified partial baseline component and a nonparametric variance function. An interesting finding is that the performance of a naive weighted version of the existing estimator could deteriorate when the smooth baseline component is badly estimated. To avoid this, we propose a family of consistent estimators and investigate their asymptotic properties. We show that the optimal semiparametric efficiency bound can be reached by a semiparametric kernel estimator in this family. Building upon our theoretical findings and heuristic arguments about the equivalence between kernel and spline smoothing, we conjecture that a weighted partial-spline estimator could also be semiparametric efficient. Properties of the proposed estimators are presented through theoretical illustration and numerical simulations.
Key Words: Double robustness; Kernel estimation; Influence function; Partial spline; Semiparametric efficiency bound
Received December 2004. Revised October 2005.