A Dichotomous Behavior of Guttman-Kaiser Criterion from Equi-Correlated Normal Population

We consider a $p$-dimensional, centered normal population such that all variables have a positive variance and any correlation coefficient between different variables is a given nonnegative constant $\rho<1$. Suppose that both the sample size $n$ and population dimension $p$ tend to infinity with $p/n \to c>0$. We prove that the limiting spectral distributions of the sample covariance matrices and the sample correlation matrices are Mar\v{c}enko-Pastur distribution of index $c$ and scale parameter $1-\rho$.By the limiting spectral distributions, we rigorously show the limiting behavior of widespread stopping rules Guttman-Kaiser criterion and cumulative-percentage-of-variation rule in PCA and EFA.As a result, we establish the following dichotomous behavior of Guttman-Kaiser criterion when both $n$ and $p$ are large, but $p/n$ is small: (1) the criterion retains a small number of variables for $\rho>0$, as suggested by Kaiser, Humphreys, and Tucker [Kaiser, H. F. (1992). On Cliff's formula, the Kaiser-Guttman rule and the number of factors. \emph{Percept. Mot. Ski.} 74]; and(2) the criterion retains $p/2$ variables for $\rho=0$, as in a simulation study [Yeomans, K. A. and Golder, P. A. (1982). The Guttman-Kaiser criterion as a predictor of the number of common factors. \emph{J. Royal Stat. Soc. Series D} 31(3)].