Asymptotically efficient estimation of linear functionals in inverse regression models

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 etal. [Klaassen, C.A.J., Lee, EA. 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 Hajek-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. I Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A., 1993, Efficient and Adaptive Estimation for Semi-parametric 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.