**On the Distribution of
the Ratio of the Largest Eigenvalue to the Trace of a Wishart Matrix**

*Written by Boaz Nadler (2011)*

Let l_{1}
denote the largest sample eigenvalue of a *p *by* p*
Wishart matrix *W* with identity covariance, computed from *n* samples. In
several problems, including testing for multiplicative components of
interaction [3] and signal detection [4], there is an interest in the distribution of the following
random variable

*U = *l_{1} / [ Trace(*W*) / p ].

In principle, as *p,n**
→ ∞, *
with *p/n→c*, after
proper centering and scaling, the asymptotic distribution of the ratio is the
same as that of the largest eigenvalue itself, namely a Tracy-Widom distribution

However, as shown in [1], this may be a poor approximation to this
distribution, in particular for small to modest values of
*p,n*. Instead, in [1] a refined
approximation was derived

We provide a Matlab function that inverts the
modified distribution for the ratio, and for a given right tail probability
α provides an approximate
threshold t(α,p,n), such that
Pr[*U* > t] ~ α.

function [t s ] = TW_trace_ratio_threshold
(p, n, beta, alpha)

p = dimension

n = number of samples,

beta = 1 or 2 for real or complex valued noise,

alpha = right tail probability, a number between 0 and 1.

Additional required routines are:

function [mu_np sigma_np] = KN_mu_sigma(n,p,beta); %
computes the centering and scaling constants

TW_beta1.mat, TW_beta2.mat % pre-computed tables of the Tracy-Widom distributions and their derivatives, using Matlab code by Prof. Folkmar Bornemann and based on [5]

**
Demo:** function demo_ratio.m compares the accuracy
of the theoretical formula to empirical simulations, as a function of number of samples
*n*, similar
to the figure in the paper [1].

[1] B. Nadler, On the
distribution of the ratio of the largest eigenvalue to the trace of a
Wishart matrix, *Journal of Multivariate Analysis,*
2011.

[2]
I.M. Johnstone, On the distribution of the largest
eigenvalues in principal components analysis, *Annals of Statistics*, vol.
29, pp. 295-327, 2001.

[3] J.R. Schott,
A note on the critical values used in stepwise tests for
multiplicative components of interaction, *Communications in Statistics -
Theory and Methods,* 1986.

[5] F. Bornemann, On the numerical evaluation of distributions in random matrix theory,
* Markov Processes and Related Fields,* 2010.