## 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.

### Other resources

• Ioana Dumitriu's MOPs package for computing with Multivariate Orthogonal Polynomials.

### 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) Computes the multivariate Gamma function of parameter beta. beta=2/alpha.

### pdfTraceWishart

Syntax: f=pdfTraceWishart(MAX,u,Sigma,m) Computes the pdf of the tr(A), where A is an m-by-n Wishart matrix with m degrees of freedom and covariance matrix Sigma. 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 A script for testing pdfTraceWishart against the results of a Monte-Carlo experiment. 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) returns the pdf of the smallest eigenvalue of an n-by-n beta-Laguerre matrix of parameter a. 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*(n-1)/2 x is a set of values on which the pdf is to be evaluated See Ph.D thesis of Ioana Dumitriu, p. 146 for theoretical background. See our paper, sec. 6.2 for an example.

### pdfMinEigBetaLaguerreTEST

Syntax: pdfMinEigBetaLaguerreTEST A script for testing pdfMinEigBetaLaguerre against the result of a Monte-Carlo experiment.

### cdfMaxEigWishart

Syntax: f=cdfMaxEigWishart(MAX) Computes the cumulative distribution function of the largest eigenvalue of an n-by-n Wishart matrix with k degrees of freedom and covariance matrix Sigma. Enter your own values for m, Sigma, and MAX. MAX must be large enough to guarantee convergence of the underlying series.

### cdfMaxEigWishartTEST

Syntax: cdfMaxEigWishartTEST A script for testing cdfMaxEigWishart against the results of a Monte-Carlo experiment. 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() Comutes the CDF of the largest eigenvalue of an n-by-n Beta-Laguerre matrix of parameter a. Enter your own values for m, Sigma, and MAX. MAX must be large enough to guarantee convergence of the underlying series.

### cdfMaxEigBetaLaguerreTEST

Syntax: cdfMaxEigBetaLaguerreTEST A script for testing cdfMaxEigBetaLaguerre against the results of a Monte-Carlo experiment. 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