Convexity of acceptance regions of multivariate tests (Q1059957)

We show that if \(\phi\) is an increasing convex function of the eigenvalues of a symmetric matrix argument, then \(Y\to \phi ((YY^ T)^{1/2})\) is a convex function. This has the consequence for the multivariate analysis of variance that statistical tests for \(\mu =0\) which are defined by increasing convex functions of the square roots of the eigenvalues of \((ZZ^ T)^{-1}YY^ T\) have acceptance regions which are convex in Y for fixed Z. Tests of this form have some good properties, including unbiasedness and power functions which are increasing in the noncentrality parameters.