Conditional Estimation of Linear Functionals

The article considers the problem of statistical estimation of linear functionals, this incorporating the isolation of a constant signal in additive noise for the case in which noise parameters such as the variance, asymmetry, etc., are known. $U$-statistics are used to construct a class of estimates $T_n$ with adaptation; it is established that they are asymptotically optimal (locally minimax) relative to a broad class of loss functions under certain general sufficient conditions. As is shown, the latter are close to being necessary in a certain sense.