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September, 1980 The Limiting Empirical Measure of Multiple Discriminant Ratios
Kenneth W. Wachter
Ann. Statist. 8(5): 937-957 (September, 1980). DOI: 10.1214/aos/1176345134


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.


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Kenneth W. Wachter. "The Limiting Empirical Measure of Multiple Discriminant Ratios." Ann. Statist. 8 (5) 937 - 957, September, 1980.


Published: September, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0473.62050
MathSciNet: MR585695
Digital Object Identifier: 10.1214/aos/1176345134

Primary: 62H30
Secondary: 60F05

Rights: Copyright © 1980 Institute of Mathematical Statistics


Vol.8 • No. 5 • September, 1980
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