The asymptotic distribution of principal component roots under local alternatives to multiple roots (Q797929)

Let \(\Sigma_ n\) be a sequence of nonrandom symmetric positive definite matrices of order p, and let \(\Gamma_ n\) be any sequence of \(p\times p\) nonrandom matrices such that \(\Gamma^ t_ n\Gamma_ n=\Sigma_ n^{-1}\). Further, let \(S_ n\) be a sequence of random symmetric positive definite matrices of order p such that \(n^{{1\over2}}\Gamma_ n(S_ n-\Sigma_ n)\Gamma^ t_ n\) are asymptotically normal and possess certain invariance properties. In this paper, the asymptotic behavior of the principal component roots of \(S_ n\) is studied under a sequence of local alternatives to multiple population roots. As an application of the result, a local power function for the test for subsphericity is studied.