We consider the usual estimator of a linear functional ofApplication of the Hájek - LeCamConvolution TheoremEun-Joo Lee

Texas Tech UniversityAbstract

the unknown function in inverse nonparametric regression

models. The unknown regression function which is the parameter

of interest, is infinite dimensional in the nonparametric

regression models. Inverse problems arise in many areas.

Examples are Wicksell's unfolding problem, computer tomography,

and radio astronomy, etc. Usually, the output is an integral

transform of the input. Therefore the transformation must be

inverted to recover the input. Because such an inversion is,

in general, unbounded, we require regularization. Since

a function in a Hilbert space has a Fourier expansion in an

orthonormal basis, we estimate an unknown input function by

estimating its Fourier coefficients. It is surprising to see

that the traditional estimator of the Fourier coefficient is

not asymptotically efficient according to H\'ajek - LeCam

convolution theorem. Since this estimator, however, is

$\sqrt n$- consistent, it can be improved. In H\'ajek the

parameter is in Euclidean space, and van der Vaart (1998)

allows an infinite dimensional parameter. Van der Vaart has

proved the convolution theorem using geometric interpretation.

We will give an independent proof in the case of pure point

spectrum.