Brenda MacGibbon - On statistical minimax estimation and principal eigenfunctions of the Laplacian

In many parametric statistical estimation problems, there is definite prior information concerning the values of the parameter vector. There may be bounds on the individual components or on a particular functional of the whole vector. Many computationally feasible estimation methods have been developed to capitalize on such information, but theoretical results have lagged behind. One approach is the worst case analysis: given some error measure, compute the maximum expected error over the restricted parameter space. The study of the resulting best or minimax risk is related here to the study of the classical Dirichlet problem and of principal eigenfunctions of the Laplacian.