On adaptive estimation of linear functionals from observations against white noise

We consider the problem of adaptive estimation of a linear functional of an unknown multivariate vector from its observations against white Gaussian noise. As a family of estimators for the functional, we use those generated by projection estimators of the unknown vector, and the main problem is to select the best estimator in this family. The goal of the paper is to explain and mathematically justify a simple statistical idea used in adaptive (i.e., observation-based) choice of the best estimator of a linear functional from a given family of estimators. We also discuss generalizations of the considered statistical model and the proposed estimation method, which allow to cover a broad class of statistical problems.