A New Approach to the Stochastic Recovery Problem

Abstract

We consider the problem of minimax estimation of linear functionals on function classes, also known as the recovery problem. We propose a new formalization of the original applied problem ---\ in both deterministic and stochastic settings. For the latter case we propose a natural probability measure on the set generated by the problem's information operator. We then provide some examples of solving recovery problems in the new framework and determine the statistical properties of the stochastic recovery problem's solution. Finally, we consider an application of the proposed approach to the problem of nonparametric minimax estimation of the regression function.