On the Bias in Eigenvalues of Sample Covariance Matrix

Abstract

Principal component analysis (PCA) is a multivariate statistical technique frequently employed in research in behavioral and social sciences, and the results of PCA are often used to approximate those of exploratory factor analysis (EFA) because the former is easier to implement. In practice, the needed number of components or factors is often determined by the size of the first few eigenvalues of the sample covariance/correlation matrix. Lawley (1956) showed that if eigenvalues of population covariance matrix are distinct, then each sample eigenvalue contains a bias of order 1/N, which is typically ignored in practice. This article further shows that, under some regulatory conditions, the order of the bias term is p/N. Thus, when p is large, the bias term is no longer negligible even when N is large.