Random Matrix Statistics
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Under Construction! Current version posted: August 18, 2005. You need the
mhg package for the functions here to work.
Installation instructions:
Download and unzip the package rms.zip.
Other resources
 Ioana Dumitriu's MOPs
package for computing with Multivariate Orthogonal Polynomials.
Function description:
mhg, mhgi, logmhg
Description: 
Computes the Hypergeometric function of one or
two matrix arguments. See here for
details.

gammam
Syntax: 
y=gammam(beta,m,a) 
Description: 
Computes the multivariate Gamma function of parameter beta.

Comments: 
beta=2/alpha.

pdfTraceWishart
Syntax: 
f=pdfTraceWishart(MAX,u,Sigma,m) 
Description: 
Computes the pdf of the tr(A), where A is an mbyn Wishart matrix
with m degrees of freedom and covariance matrix Sigma.

Comments: 
 See Muirhead, 1978, p. 341 for theoretical background.
 u is a vector, e.g., u=[0:0.1:30]; the function returns f(u),
where f is the pdf of tr(A)
 Although in practice Sigma is a matrix, only its eigenvalues factor in
the computation, so the parameter Sigma is simply a vector of its
eigenvalues.
 A large enough
value of MAX should be supplied to guarantee convergence of the series.
 Run pdfTraceWishartTEST to see an example.

pdfTraceWishartTEST
Syntax: 
pdfTraceWishartTEST 
Description: 
A script for testing pdfTraceWishart against the results of a
MonteCarlo experiment. 
Comments: 
Enter your own values for
m, Sigma, and MAX. MAX must be large enough to guarantee
convergence of the underlying series.

pdfMinEigBetaLaguerre
Syntax: 
f=pdfMinEigBetaLaguerre(M,beta,n,a,x) 
Description: 
returns the pdf of the smallest eigenvalue of an nbyn
betaLaguerre matrix of parameter a.

Arguments: 
 M is an integer, indicates the depth of the truncation, larger
values will yield more accurate result, but will also take more time.
 beta is positive and equals 2/alpha
 n is the size of the matrix
 a is a parameter, we must have a>beta*(n1)/2
 x is a set of values on which the pdf is to be evaluated

Comments: 

pdfMinEigBetaLaguerreTEST
Syntax: 
pdfMinEigBetaLaguerreTEST 
Description: 
A script for testing pdfMinEigBetaLaguerre against the
result of a MonteCarlo experiment.

cdfMaxEigWishart
Syntax: 
f=cdfMaxEigWishart(MAX) 
Description: 
Computes the cumulative distribution function of the largest
eigenvalue of an nbyn Wishart matrix with k degrees of freedom and
covariance matrix Sigma.

Comments: 
Enter your own values for
m, Sigma, and MAX. MAX must be large enough to guarantee
convergence of the underlying series.

cdfMaxEigWishartTEST
Syntax: 
cdfMaxEigWishartTEST 
Description: 
A script for testing cdfMaxEigWishart against the results of a
MonteCarlo experiment. 
Comments: 
Enter your own values for
m, Sigma, and MAX. MAX must be large enough to guarantee
convergence of the underlying series.

cdfMaxEigBetaLaguerre
Syntax: 
f=cdfMaxEigBetaLaguerre() 
Description: 
Comutes the CDF of the largest eigenvalue of an nbyn
BetaLaguerre matrix of parameter a.

Comments: 
Enter your own values for
m, Sigma, and MAX. MAX must be large enough to guarantee
convergence of the underlying series.

cdfMaxEigBetaLaguerreTEST
Syntax: 
cdfMaxEigBetaLaguerreTEST 
Description: 
A script for testing cdfMaxEigBetaLaguerre against the results of
a MonteCarlo experiment.

Comments: 
Enter your own values for
m, Sigma, and MAX. MAX must be large enough to guarantee
convergence of the underlying series.

Plamen Koev Department of Mathematics, Massachusetts Institute
of Technology
"firstname"@math.mit.edu

