Asymptotic distribution of the LR statistic for equality of the smallest eigenvalues in high-dimensional principal component analysis

Abstract

This paper deals with the distribution of the LR statistic for testing the hypothesis that the smallest eigenvalues of a covariance matrix are equal. We derive an asymptotic null distribution of the LR statistic when the dimension p and the sample size N approach infinity, while the ratio p / N converging on a finite nonzero limit c ∈ ( 0 , 1 ) . Numerical simulations revealed that our approximation is more accurate than the classical chi-square-type approximation as p increases in value.