Application of the Hájek - LeCam
Convolution Theorem

Eun-Joo Lee
Texas Tech  University

Abstract

We consider the usual estimator of a linear functional of
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.