The asymptotic distribution of principal component roots under local alternatives to multiple roots. (English) Zbl 0546.62007

Tyler, David E.

Ann. Stat. 11, 1232-1242 (1983).

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.

Reviewer: K.Yoshihara

MSC:

62E20	Asymptotic distribution theory in statistics
62H25	Factor analysis and principal components; correspondence analysis
62H10	Multivariate distribution of statistics
15B52	Random matrices (algebraic aspects)

Keywords:

affine-invariant M-estimates of scatter; elliptical distributions; spherically invariant random matrices; principal component roots; local alternatives to multiple population roots; local power function; test for subsphericity