multivariate_student_pdf

 Evaluates the density of a multivariate student, with expectation Mean,
 variance Sigma_upper_chol'*Sigma_upper_chol and degrees of freedom df, at X.

 INPUTS 

    X                  [double]    1*n vector        
    Mean               [double]    1*n vector, expectation of the multivariate random variable.
    Sigma_upper_chol   [double]    n*n matrix, upper triangular Cholesky decomposition of Sigma (the "covariance matrix").
    df                 [integer]   degrees of freedom.
    
 OUTPUTS 
    density            [double]    density. 
        
 SPECIAL REQUIREMENTS

0001 function density = multivariate_student_pdf(X,Mean,Sigma_upper_chol,df);
0002 % Evaluates the density of a multivariate student, with expectation Mean,
0003 % variance Sigma_upper_chol'*Sigma_upper_chol and degrees of freedom df, at X.
0004 %
0005 % INPUTS
0006 %
0007 %    X                  [double]    1*n vector
0008 %    Mean               [double]    1*n vector, expectation of the multivariate random variable.
0009 %    Sigma_upper_chol   [double]    n*n matrix, upper triangular Cholesky decomposition of Sigma (the "covariance matrix").
0010 %    df                 [integer]   degrees of freedom.
0011 %
0012 % OUTPUTS
0013 %    density            [double]    density.
0014 %
0015 % SPECIAL REQUIREMENTS
0016 
0017 % Copyright (C) 2003-2009 Dynare Team
0018 %
0019 % This file is part of Dynare.
0020 %
0021 % Dynare is free software: you can redistribute it and/or modify
0022 % it under the terms of the GNU General Public License as published by
0023 % the Free Software Foundation, either version 3 of the License, or
0024 % (at your option) any later version.
0025 %
0026 % Dynare is distributed in the hope that it will be useful,
0027 % but WITHOUT ANY WARRANTY; without even the implied warranty of
0028 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
0029 % GNU General Public License for more details.
0030 %
0031 % You should have received a copy of the GNU General Public License
0032 % along with Dynare.  If not, see <http://www.gnu.org/licenses/>.
0033 nn = length(X);
0034 t1 = gamma( .5*(nn+df) ) / ( gamma( .5*nn ) * (df*pi)^(.5*nn) ) ;
0035 t2 = t1 / prod(diag(Sigma_upper_chol)) ;
0036 density = t2 / ( 1 + (X-Mean)*(Sigma_upper_chol\(transpose(Sigma_upper_chol)\transpose(X-Mean)))/df )^(.5*(nn+df));