Asymptotically efficient estimation of linear functionals in inverse regression models

Abstract

In this paper, we will discuss a procedure to improve the usual estimator of a linear functional of the unknown regression function in inverse non-parametric regression models. In Klaassen et al. [Klaassen, C.A.J., Lee, E.-J. and Ruymgaart, F.H., 2001, On efficiency of indirect estimation of nonparametric regression functions. In: M.A.G. Viana and D.St.P. Richards (Eds) Algebraic Methods in Statistics and Probability. Contemporary Mathematics, Vol. 287 (Providence, Rhode Island: American Mathematical Society), pp. 173–184.], it has been proved that this traditional estimator is not asymptotically efficient (in the sense of the Hájek–LeCam convolution theorem) except, possibly, when the error distribution is normal. As this estimator, however, is still root-n consistent, a procedure in Bickel et al. [Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A., 1993, Efficient and Adaptive Estimation for Semiparametric Models (Baltimore: Johns Hopkins University Press).] applies to construct a modification which is asymptotically efficient. A self-contained proof of the asymptotic efficiency is included. In addition, some simulations are performed.