Estimating linear functionals of indirectly observed input functionals

Abstract

We consider the usual estimator of a linear functional of the unknown input function in indirect nonparametric regression models. The unknown regression function which is the parameter of interest, is infinite dimensional. Since the output is an integral transform of the input, this transformation must be inverted to recover the input. Because such an inversion is, in general, unbounded, regularization of the inverse will be required. Since a function in a separable Hilbert space has a Fourier expansion in an orthonormal basis the Fourier coefficients will be estimated in order to recover the input. Regularization of the inverse boils down to tapering the expansion with the estimated Fourier coefficients, which would otherwise not converge. In any case this shows that estimating Fourier coefficients and linear functionals in general is an important issue. It is surprising to see that the traditional estimator of the Fourier coefficients is not asymptotically efficient according to the Hajek-LeCam convoluteion theorem. Since this estimator, however, is -y/n-consistent, it can be improved in an asymptotic sense. A simulation study is included to establish the practical effect of this asymptotic result. In this dissertation, the theory is presented in a self-contained manner. This means that a complete derivation of theorems like Hajek's convolution theorem and the theorem on possible improvement of n-consistent estimators will be given.