MINIMAX ESTIMATION OF LINEAR FUNCTIONALS, PARTICULARLY IN NONPARAMETRIC REGRESSION AND POSITRON EMISSION TOMOGRAPHY

Often it is required to estimate a linear functional of a function f given observations of some kind from f. We consider the linear minimax estimation of a bounded linear functional T with respect to the mean square error loss function over a certain class of functions F defined by means of a semi-norm, which in practice usually provides some restriction on roughness. We review the remarkable result essentially due to Speckman (1979) that the linear minimax estimate of T(f) is T(($) over cap f), where ($) over cap f is a type of penalized least squares estimate of f. Hence the amount and method of smoothing carried out in the minimax estimation of T(f) does not depend on the functional T under consideration. The function f* in F for which the maximum of the loss function is attained does, however, depend on T. We discuss the result in the case of nonparametric regression and positron emission tomography (PET). In the former case we see a surprising consequence of the dependence of f* on T. In the latter case we show how the result can be applied to PET and illustrate it numerically, highlighting certain interesting features of the PET process.