Some extensions of the Kantorovich inequality and statistical applications

Abstract

Kantorovich gave an upper bound to the product of two quadratic forms, (X'AX) (X'A^-1X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X'AX| |X'A^-1X| where X is n × k matrix such that X'X = I_k. In this paper we determine the bounds for the traces and determinants of matrices of the type X'AYY'A^-1X, X'B²X(X'BCX)^-1 X'C²X(X'BCX)^-1 where X and Y are n × k matrices such that X'X = Y'Y = I_k and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.