Linear minimax estimation for inadequate models in L 2Metric

Abstract

Let Q be a compact in the Euclidean space E^p , μ (dx) a measure defined on Qand F,G – two linear finite-dimensional subspaces of a -space of continuous functions on Q. Let f(x) be an unknown function from F and let the -distance of f(x).from G be known not to exceed a given constant α². Given the measurements of f x with uncorrelated errors on a set of points in Q, the -norm optimal linear estimation of f ∈ F as a function of G is formulated as a minimax problem. Where the maximum is taken over all f ∈ F with . This estimation problem is reduced to the matrix problem where λ(X^TX) is the maximal eigenvalue of the matrix X^TX. Basing on this matrix problem the existence of the solution, some properties, bounds and suboptimal solutions for the initial estimation problem are obtained.