On minimax estimating Hilbert random elements
Konstantin Siemenikhin
The problem of designing optimal algorithms for estimating random elements has received earlier considerable attention. Basically, the works devoted to the infinite-dimensional estimation problems deal with linear procedures and do not discuss the efficiency of nonlinear estimates. On the other hand, the nonlinear optimal estimation algorithms are essentially based on using the true distribution of the random elements involved. Nevertheless, these obstacles of optimal methods can be overcome by means a minimax approach. Actually, even though the class of estimators contains all nonlinear transformations, the minimax estimate turns to be linear under very broad assumptions. In the finite-dimensional case, this result holds whenever the covariances of the model parameters are supposed to belong to a compact set. Furthermore, it turns out that for the uncertainty set under consideration the least favorable distribution is Gaussian. In this paper, the analogous results are proved for the infinite-dimensional model.
Using the technique of dual optimization we provide the sufficient conditions for the minimax estimate to be defined analytically via a solution of the dual optimization problem. Thus, if the least favorable covariance (i.e., the solution of the dual problem) is found, the minimax estimate should be designed as the optimal one. For numerical calculation of the minimax estimate we present the recursive algorithm which takes into account only finite-dimensional transformations of the observed random element.