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 nonparametric regression models. In Klaassen, Lee, and Ruymgaart (2001) it has been proved that this traditional estimator is not asymptotically efficient (in the sense of the H\'{a}jek - Le Cam convolution theorem) except, possibly, when the error distribution is normal. Since this estimator, however, is still root-n consistent a procedure in Bickel, Klaassen, Ritov, and Wellner (1993) applies to construct a modification which is asymptotically efficient. A self-contained proof of the asymptotic efficiency is included.