The Limiting Empirical Measure of Multiple Discriminant Ratios

Abstract

Consider the positive roots of the determinental equation $\det|YJY^\ast - x^2YY^\ast| = 0$ for a $p(n)$ by $n$ sample matrix of independent unit Gaussians $Y$ with transpose $Y^\ast$ and a projection matrix $J$ of rank $m(n).$ We prove that the empirical measure of these roots converges in probability to a nonrandom limit $F$ as $p(n), m(n),$ and $n$ go to infinity with $p(n)/n \rightarrow \beta$ and $m(n)/n \rightarrow \mu$ in $(0, 1).$ Along with possible atoms at zero and one, $F$ has a density proportional to $((x - A)(x + A)(B - x)(B + x))^\frac{1}{2}/\lbrack x(1 - x)(1 + x) \rbrack$ between $A = |(\mu - \mu \beta)^\frac{1}{2} - (\beta - \mu \beta)^\frac{1}{2}|$ and $B = |(\mu - \mu\beta)^\frac{1}{2} + (\beta - \mu \beta)^\frac{1}{2}|.$ On the basis of this result, tables of quantiles are given for probability plotting of multiple discriminant ratios, canonical correlations, and eigenvalues arising in MANOVA under the usual null hypotheses when the dimension and degree of freedom parameters are large.