## The Annals of Statistics

### The Limiting Empirical Measure of Multiple Discriminant Ratios

Kenneth W. Wachter

#### 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.

#### Article information

Source
Ann. Statist., Volume 8, Number 5 (1980), 937-957.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345134

Digital Object Identifier
doi:10.1214/aos/1176345134

Mathematical Reviews number (MathSciNet)
MR585695

Zentralblatt MATH identifier
0473.62050

JSTOR
links.jstor.org

#### Citation

Wachter, Kenneth W. The Limiting Empirical Measure of Multiple Discriminant Ratios. Ann. Statist. 8 (1980), no. 5, 937--957. doi:10.1214/aos/1176345134. https://projecteuclid.org/euclid.aos/1176345134