Asymptotic Theory for Principal Component Analysis

TLDR

The asymptotic distribution of eigenvalues and eigenvectors of a sample covariance matrix is derived for multivariate normal data with arbitrary multiplicity of eigenvalues, where eigenvectors represent principal components and eigenvalues represent their variances. The study develops tests for equality of population eigenvalues and provides confidence intervals for equal eigenvalues to evaluate the importance of principal components. A similar analysis is performed for correlation matrices. The author is (Author).

Abstract

Abstract : The asymptotic distribution of the characteristic roots and (normalized) vectors of a sample covariance matrix is given when the observations are from a multivariate normal distribution whose covariance matrix has characteristic roots of arbitrary multiplicity. The elements of each characteristic vector are the coefficients of a principal component (with sum of squares of coefficients being unity), and the corresponding characteristic root is the variance of the principal component. Tests of hypotheses of equality of population roots are treated, 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. (Author)