On Certain Characteristics of the Distribution of the Latent Roots of a Symmetric Random Matrix Under General Conditions

Abstract

Under certain conditions, to be specified in Theorems 2 and 4, the latent roots of the symmetric random matrix $F$ with $\varepsilon F = \Phi$ are biased estimators of the latent roots of $\Phi$; the smallest (largest) root is negatively (positively) biased. Here bias includes both expectation-bias and median-bias. Further properties of the distribution of the latent roots are given, among them some relations between covariances of the latent roots, covariances of elements of $F$, and the amounts of expectation-bias of the latent roots. Also, a sufficient condition is given for a certain type of symmetry in the joint distribution of the latent roots. For applications of the theory presented in this paper to the theory of response surface estimation see van der Vaart [9].