Asymptotics of Pearson-Hotelling Principal-Component Vectors of Sample Variance and Correlation Matrices

Abstract

Asymptotic normal distributions of eigenvalue-normed eigenvectors (Pearson-Hotelling principal-component vectors) of sample variance and correlation matrices are derived. Each distribution follows from the asymptotic distribution of the whole matrix of eigenvectors, also obtained in the paper. The results are presented for the general case, where the existence of fourth-order finite population moments is assumed, and also specified for normal and elliptical populations. Population variance and correlation matrices are supposed to be nonsingular and without multiple eigenvalues. A comparison with unit-length eigenvectors (Anderson principal-component vectors) is being made. Special attention is being paid to asymptotic variances of eigenvectors of both sample variance and correlation matrices in the two approaches. It is shown that the asymptotic variance matrices of principal-component vectors for sample variance and correlation matrices are singular in the Anderson approach. In the Pearson-Hotelling approach, however, the asymptotic variance of eigenvectors of the sample variance matrix is nonsingular, whereas in the sample correlation matrix case the asymptotic variance matrix of eigenvectors is nonsingular under certain conditions.