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.
Keywords
MathematicsCombinatoricsZero (linguistics)DiscriminantBETA (programming language)Dimension (graph theory)Rank (graph theory)Statistics
Related Publications
Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results
Let ${M}$ be a compact Riemannian submanifold of ${{\\bf R}^m}$ of dimension\n$\\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$\nwith uniform dist...
A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables
Let $(X_n)_{n \\in \\mathscr{X J}}$ be a sequence of r.v.'s with $E X_n = 0, E(\\sum^n_{i = 1} X_i)^2/n \\rightarrow \\sigma^2 > 0, \\sup_{n,m}E(\\sum^{m + n}_{i = m + 1} X_i...
A Limit Theorem for the Norm of Random Matrices
This paper establishes an almost sure limit for the operator norm of rectangular random matrices: Suppose $\\{v_{ij}\\}i = 1,2, \\cdots, j = 1,2, \\cdots$ are zero mean i.i.d. r...
Limit of the Smallest Eigenvalue of a Large Dimensional Sample Covariance Matrix
In this paper, the authors show that the smallest (if $p \\leq n$) or the $(p - n + 1)$-th smallest (if $p > n$) eigenvalue of a sample covariance matrix of the form $(1/n)XX...
Asymptotically Most Powerful Rank-Order Tests
Having observed $X_i = \\alpha + \\beta c_i + \\sigma Y_i$, we test the hypothesis $\\beta = 0$ against the alternative $\\beta > 0$. We suppose that the square root of the p...
Publication Info
- Year
- 1980
- Type
- article
- Volume
- 8
- Issue
- 5
- Citations
- 102
- Access
- Closed
External Links
Social Impact
Altmetric
PlumX Metrics
Social media, news, blog, policy document mentions
Citation Metrics
102
OpenAlex
Cite This
Kenneth W. Wachter
(1980).
The Limiting Empirical Measure of Multiple Discriminant Ratios.
The Annals of Statistics
, 8
(5)
.
https://doi.org/10.1214/aos/1176345134
Identifiers
- DOI
- 10.1214/aos/1176345134