  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.
