Statistics: mvtcdf

### 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");