On the bias of functions of characteristic roots of a random matrix

Let Z = (zij) be arandomp xp symmetricmatrixwith EZ = A, i.e. Ezij = aj (i,j= 1,... ,p). An application to the theory of response surface estimation led van der Vaart (1961) to consider the expectation-bias and median-bias of the characteristic roots of Z as estimators of the characteristic roots of A. Denote the (real) characteristic roots of a matrix X by A(X) with the ordering A1(X) > ... > Ap(X). Van der Vaart proved that if Z is a symmetric matrix, then EA1(Z) > A1(A) and EAp(Z) O} = P{trC(Z -A) A1(A)} > 1 and P{Ap(Z) 2. If absolute continuity is assumed, these bias inequalities become strict. The purpose of this paper is to show how to generate a wide class of inequalities between EA(Z) and A(EZ). In particular, some of the results of van der Vaart (1961) can be strengthened by a weakening of the assumption that Z be a symmetric matrix. When Z is symmetric, the expectation-bias and median-bias may also be obtained for partial sums of the roots, and when Z is positive definite, the expectation-bias is obtained for more general functions of the roots. Expectation-bias for the roots of the determinantal equation IZl Z21 = 0 is presented; of particular interest is the determinantal equation for the canonical correlations. Inequalities for the median-bias of certain linear combinations of the roots are also developed. These inequalities are obtained as direct consequences of known results concerning the convexity of scalar functions of a matrix. An extension of Jensen's inequality to convex matrix functions using the Loewner ordering for matrices is also given; this extension then serves as a source of other inequalities.