Multivariate *t* cumulative distribution
function

`y = mvtcdf(X,C,DF)`

y = mvtcdf(xl,xu,C,DF)

[y,err] = mvtcdf(...)

[...] = mvntdf(...,options)

`y = mvtcdf(X,C,DF)`

returns
the cumulative probability of the multivariate *t* distribution
with correlation parameters `C`

and degrees of freedom `DF`

,
evaluated at each row of `X`

. Rows of the *n*-by-*d* matrix `X`

correspond
to observations or points, and columns correspond to variables or
coordinates. `y`

is an `n`

-by-`1`

vector.

`C`

is a symmetric, positive definite, *d*-by-*d* matrix,
typically a correlation matrix. If its diagonal elements are not 1, `mvtcdf`

scales `C`

to
correlation form. `mvtcdf`

does not rescale `X`

. `DF`

is
a scalar, or a vector with *n* elements.

The multivariate *t* cumulative probability
at `X`

is defined as the probability that a random
vector `T`

, distributed as multivariate *t*,
will fall within the semi-infinite rectangle with upper limits defined
by `X`

, i.e., `Pr{T(1)`

≤`X(1),T(2)`

≤`X(2),...T(`

≤*d*)`X(`

.*d*)}

`y = mvtcdf(xl,xu,C,DF)`

returns
the multivariate *t* cumulative probability evaluated
over the rectangle with lower and upper limits defined by `xl`

and `xu`

,
respectively.

`[y,err] = mvtcdf(...)`

returns
an estimate of the error in `y`

. For bivariate and
trivariate distributions, `mvtcdf`

uses adaptive
quadrature on a transformation of the *t* density,
based on methods developed by Genz, as described in the references.
The default absolute error tolerance for these cases is `1e-8`

.
For four or more dimensions, `mvtcdf`

uses a quasi-Monte
Carlo integration algorithm based on methods developed by Genz and
Bretz, as described in the references. The default absolute error
tolerance for these cases is `1e-4`

.

`[...] = mvntdf(...,options)`

specifies
control parameters for the numerical integration used to compute `y`

.
This argument can be created by a call to `statset`

.
Choices of `statset`

parameters are:

`'TolFun'`

— Maximum absolute error tolerance. Default is`1e-8`

when*d*< 4, or`1e-4`

when*d*≥ 4.`'MaxFunEvals'`

— Maximum number of integrand evaluations allowed when*d*≥ 4. Default is`1e7`

.`'MaxFunEvals'`

is ignored when*d*< 4.`'Display'`

— Level of display output. Choices are`'off'`

(the default),`'iter'`

, and`'final'`

.`'Display'`

is ignored when*d*< 4.

[1] Genz, A. "Numerical Computation
of Rectangular Bivariate and Trivariate Normal and t Probabilities." *Statistics
and Computing*. Vol. 14, No. 3, 2004, pp. 251–260.

[2] Genz, A., and F. Bretz. "Numerical
Computation of Multivariate t Probabilities with Application to Power
Calculation of Multiple Contrasts." *Journal of Statistical
Computation and Simulation*. Vol. 63, 1999, pp. 361–378.

[3] Genz, A., and F. Bretz. "Comparison
of Methods for the Computation of Multivariate t Probabilities." *Journal
of Computational and Graphical Statistics*. Vol. 11, No.
4, 2002, pp. 950–971.

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