Minimax estimation of linear functionals, particularly in nonparametric regression and positron emission tomography (Q1965985)

Let \(H\) be a vector space of real-valued functions equipped with a semi-norm \(\|\cdot\|\) and \(T\colon H \to R\) be a bounded linear functional. This paper deals with minimax estimates of \(T(f)\) with respect to the mean square error loss function over a certain class of functions \(F\) (\(F =\{f \colon \|f\|\leq c\) for example). The authors give a new proof of an unpublished result by Speckman (1979) which states that the linear minimax estimate of \(T(f)\) is \(T(\hat f),\) where \(\hat f\) is a penalized least squares estimate of \(f.\) Formulas for computing \(T(\hat f)\) and \(\hat f\) are given. Minimax estimates are found in the case of a nonparametric regression model. Linear functionals of positron emission tomography images are estimated, too.