Unimprovable solution to systems of empirical linear algebraic equations

Abstract

An optimum solution, free from degeneration, is found for a system of linear algebraic equations with empirical coefficients and right-hand sides. The quadratic risk of estimators of the unknown solution vector is minimized over a class of linear systems with given square norm of the coefficient matrix and length of the vector on the right-hand side. Empirical coefficients and the right-hand sides are assumed to be independent and normal with known variance. It is found that the optimal estimator has the form of a regularized minimum square solution with an extension multiple. A simple formula is derived showing explicitly the dependence of the minimal risk on parameters.