--- /dev/null
+## Copyright (C) 2008 Arno Onken <asnelt@asnelt.org>
+##
+## This program is free software; you can redistribute it and/or modify it under
+## the terms of the GNU General Public License as published by the Free Software
+## Foundation; either version 3 of the License, or (at your option) any later
+## version.
+##
+## This program is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
+## details.
+##
+## You should have received a copy of the GNU General Public License along with
+## this program; if not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{p} =} mvtcdf (@var{x}, @var{sigma}, @var{nu})
+## @deftypefnx {Function File} {} mvtcdf (@var{a}, @var{x}, @var{sigma}, @var{nu})
+## @deftypefnx {Function File} {[@var{p}, @var{err}] =} mvtcdf (@dots{})
+## Compute the cumulative distribution function of the multivariate
+## Student's t distribution.
+##
+## @subheading Arguments
+##
+## @itemize @bullet
+## @item
+## @var{x} is the upper limit for integration where each row corresponds
+## to an observation.
+##
+## @item
+## @var{sigma} is the correlation matrix.
+##
+## @item
+## @var{nu} is the degrees of freedom.
+##
+## @item
+## @var{a} is the lower limit for integration where each row corresponds
+## to an observation. @var{a} must have the same size as @var{x}.
+## @end itemize
+##
+## @subheading Return values
+##
+## @itemize @bullet
+## @item
+## @var{p} is the cumulative distribution at each row of @var{x} and
+## @var{a}.
+##
+## @item
+## @var{err} is the estimated error.
+## @end itemize
+##
+## @subheading Examples
+##
+## @example
+## @group
+## x = [1 2];
+## sigma = [1.0 0.5; 0.5 1.0];
+## nu = 4;
+## p = mvtcdf (x, sigma, nu)
+## @end group
+##
+## @group
+## a = [-inf 0];
+## p = mvtcdf (a, x, sigma, nu)
+## @end group
+## @end example
+##
+## @subheading References
+##
+## @enumerate
+## @item
+## Alan Genz and Frank Bretz. Numerical Computation of Multivariate
+## t-Probabilities with Application to Power Calculation of Multiple
+## Constrasts. @cite{Journal of Statistical Computation and Simulation},
+## 63, pages 361-378, 1999.
+## @end enumerate
+## @end deftypefn
+
+## Author: Arno Onken <asnelt@asnelt.org>
+## Description: CDF of the multivariate Student's t distribution
+
+function [p, err] = mvtcdf (varargin)
+
+ # Monte-Carlo confidence factor for the standard error: 99 %
+ gamma = 2.5;
+ # Tolerance
+ err_eps = 1e-3;
+
+ if (length (varargin) == 3)
+ x = varargin{1};
+ sigma = varargin{2};
+ nu = varargin{3};
+ a = -Inf .* ones (size (x));
+ elseif (length (varargin) == 4)
+ a = varargin{1};
+ x = varargin{2};
+ sigma = varargin{3};
+ nu = varargin{4};
+ else
+ print_usage ();
+ endif
+
+ # Dimension
+ q = size (sigma, 1);
+ cases = size (x, 1);
+
+ # Check parameters
+ if (size (x, 2) != q)
+ error ("mvtcdf: x must have the same number of columns as sigma");
+ endif
+
+ if (any (size (x) != size (a)))
+ error ("mvtcdf: a must have the same size as x");
+ endif
+
+ if (! isscalar (nu) && (! isvector (nu) || length (nu) != cases))
+ error ("mvtcdf: nu must be a scalar or a vector with the same number of rows as x");
+ endif
+
+ # Convert to correlation matrix if necessary
+ if (any (diag (sigma) != 1))
+ svar = repmat (diag (sigma), 1, q);
+ sigma = sigma ./ sqrt (svar .* svar');
+ endif
+ if (q < 1 || size (sigma, 2) != q || any (any (sigma != sigma')) || min (eig (sigma)) <= 0)
+ error ("mvtcdf: sigma must be nonempty symmetric positive definite");
+ endif
+
+ nu = nu(:);
+ c = chol (sigma)';
+
+ # Number of integral transformations
+ n = 1;
+
+ p = zeros (cases, 1);
+ varsum = zeros (cases, 1);
+
+ err = ones (cases, 1) .* err_eps;
+ # Apply crude Monte-Carlo estimation
+ while any (err >= err_eps)
+ # Sample from q-1 dimensional unit hypercube
+ w = rand (cases, q - 1);
+
+ # Transformation of the multivariate t-integral
+ dvev = tcdf ([a(:, 1) / c(1, 1), x(:, 1) / c(1, 1)], nu);
+ dv = dvev(:, 1);
+ ev = dvev(:, 2);
+ fv = ev - dv;
+ y = zeros (cases, q - 1);
+ for i = 1:(q - 1)
+ y(:, i) = tinv (dv + w(:, i) .* (ev - dv), nu + i - 1) .* sqrt ((nu + sum (y(:, 1:(i-1)) .^ 2, 2)) ./ (nu + i - 1));
+ tf = (sqrt ((nu + i) ./ (nu + sum (y(:, 1:i) .^ 2, 2)))) ./ c(i + 1, i + 1);
+ dvev = tcdf ([(a(:, i + 1) - c(i + 1, 1:i) .* y(:, 1:i)) .* tf, (x(:, i + 1) - c(i + 1, 1:i) .* y(:, 1:i)) .* tf], nu + i);
+ dv = dvev(:, 1);
+ ev = dvev(:, 2);
+ fv = (ev - dv) .* fv;
+ endfor
+
+ n++;
+ # Estimate standard error
+ varsum += (n - 1) .* ((fv - p) .^ 2) ./ n;
+ err = gamma .* sqrt (varsum ./ (n .* (n - 1)));
+ p += (fv - p) ./ n;
+ endwhile
+
+endfunction
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