Asymptotic Theory for Principal Component Analysis

Abstract

The asymptotic distribution of the characteristic roots and (normalized)vectors of a sample covariance matrix is given when the observationsare from a multivariate normal distribution whose covariance matrixhas characteristic roots of arbitrary multiplicity. The elementsof each characteristic vector are the coefficients of a principalcomponent (with sum of squares of coefficients being unity), andthe corresponding characteristic root is the variance of the principalcomponent. Tests of hypotheses of equality of population roots aretreated, and confidence intervals for assumed equal roots are given;these are useful in assessing the importance of principal components.A similar study for correlation matrices is considered.