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Function Reference: mvtcdf

statistics: p = mvtcdf (x, rho, df)
statistics: p = mvncdf (x_lo, x_up, rho, df)
statistics: p = mvncdf (…, options)
statistics: [p, err] = mvncdf (…)

Multivariate Student’s t cumulative distribution function (CDF).

p = mvtcdf (x, rho, df) returns the cumulative probability of the multivariate student’s t distribution with correlation parameters rho and degrees of freedom df, evaluated at each row of x. The rows of the N×D matrix x correspond to sample observations and its columns correspond to variables or coordinates. The return argument p is a column vector with the same number of rows as in x.

rho is a symmetric, positive definite, D×D correlation matrix. dF is a scalar or a vector with N elements.

Note: mvtcdf computes the CDF for the standard multivariate Student’s t distribution, centered at the origin, with no scale parameters. If rho is a covariance matrix, i.e. diag(rho) is not all ones, mvtcdf rescales rho to transform it to a correlation matrix. mvtcdf does not rescale x, though.

The multivariate Student’s t cumulative probability at x is defined as the probability that a random vector T, distributed as multivariate normal, will fall within the semi-infinite rectangle with upper limits defined by x.

  • PrT(1)<=X(1), T(2)<=X(2), ... T(D)<=X(D).

p = mvtcdf (x_lo, x_hi, rho, df) returns the multivariate Student’s t cumulative probability evaluated over the rectangle (hyper-rectangle for multivariate data in x) with lower and upper limits defined by x_lo and x_hi, respectively.

[p, err] = mvtcdf (…) also returns an error estimate err in p.

p = mvtcdf (…, options) specifies the structure, which controls specific parameters for the numerical integration used to compute p. The required fieds are:

"TolFun"Maximum absolute error tolerance. Default is 1e-8 for D < 4, or 1e-4 for D >= 4.
"MaxFunEvals"Maximum number of integrand evaluations when D >= 4. Default is 1e7. Ignored when D < 4.
"Display"Display options. Choices are "off" (default), "iter", which shows the probability and estimated error at each repetition, and "final", which shows the final probability and related error after the integrand has converged successfully. Ignored when D < 4.

See also: bvtcdf, mvtpdf, mvtrnd, mvtcdfqmc

Source Code: mvtcdf

Example: 1

  

 ## Compute the cdf of a multivariate Student's t distribution with
 ## correlation parameters rho = [1, 0.4; 0.4, 1] and 2 degrees of freedom.

 rho = [1, 0.4; 0.4, 1];
 df = 2;
 [X1, X2] = meshgrid (linspace (-2, 2, 25)', linspace (-2, 2, 25)');
 X = [X1(:), X2(:)];
 p = mvtcdf (X, rho, df);
 surf (X1, X2, reshape (p, 25, 25));
 title ("Bivariate Student's t cummulative distribution function");
plotted figure