# cdfMvte¶

## Purpose¶

Computes multivariate Student’s t cumulative distribution function with error management.

## Format¶

{ y, err, retcode } = cdfMvte(ctl, x, corr, nonc, df)
Parameters:
• ctl (struct) –

instance of a cdfmControl structure with members

 ctl.maxEvaluations scalar, maximum number of evaluations. ctl.absErrorTolerance scalar, absolute error tolerance. ctl.relErrorTolerance scalar, tolerance.
• x (NxK matrix) – Lower limits at which to evaluate the Student’s t cumulative distribution function. If x has more than one row, each row will be treated as a separate set of upper limits. K is the dimension of the multivariate Student’s t distribution. N is the number of MVT cdf integrals.
• corr (KxK matrix) – correlation matrix.
• nonc (Kx1 vector) – noncentralities.
• df (scalar) – degrees of freedom.
Returns:
• p (N x 1 vector) – Each element in p is the cumulative distribution function of the multivariate Student’s t distribution for the corresponding elements in x.
• err (Nx1 vector) – estimates of absolute error.
• retcode (Nx1 vector) –

return codes.

 0 normal completion with $$err < ctl.absErrorTolerance$$. 1 $$err > ctl.absErrorTolerance$$ and ctl.maxEvaluations exceeded; increase ctl.maxEvaluations to decrease error. 2 $$K > 100$$ or $$K < 1$$. 3 R not positive semi-definite. missing R not properly defined.

## Examples¶

### Uncorrelated variables¶

// Upper limits of integration for K dimensional multivariate Student's t distribution
x = { 0  0 };

/*
** Identity matrix, indicates
** zero correlation between variables
*/
corr = { 1 0,
0 1 };

// Define non-centrality vector
nonc  = { 0, 0 };

// Define degree of freedom
df  = 3;

// Define control structure
struct cdfmControl ctl;
ctl = cdfmControlCreate();

/*
** Calculate cumulative probability of
** both variables being ≤ 0
*/
{ p, err, retcode } = cdfMvte(ctl, x, corr, nonc, df );

// Calculate joint probablity of two
// variables with zero correlation,
// both, being ≤ 0
p2 = (1 - cdftc(0, df)) .* (1- cdftc(0, df));


After the above code, both p and p2 should be equal to 0.25.

$T = P(-\infty < X_1 \leq 0 \text{ and } - \infty < X_2 \leq 0) \approx 0.25.$

### Compute the multivariate student’s t cdf at 3 separate pairs of upper limits¶

/*
** Upper limits of integration
** x1 ≤ -1 and x2 ≤ -1.1
** x1 ≤ 0 and x2 ≤ 0.1
** x1 ≤ 1 and x2 ≤ 1.1
*/
x = {  -1   -1.1,
0    0.1,
1    1.1 };

// Correlation matrix
corr = {   1 0.31,
0.31    1 };

// Define non-centrality vector
nonc  = {0, 0};

// Define degree of freedom
df  = 3;

// Define control structure
struct cdfmControl ctl;
ctl = cdfmControlCreate();

/*
** Calculate cumulative probability of
** each pair of upper limits
*/
{p, err, retcode}  = cdfMvte(ctl, x, corr, nonc, df);


After the above code, p should equal:

0.06752203
0.31824308
0.69617932


which means that:

$\begin{split}P(x_1 \leq -1 \text{ and } x_2 \leq -1.1) = 0.0675\\ P(x_1 \leq +0 \text{ and } x_2 \leq +0.1) = 0.3182\\ P(x_1 \leq 1 \text{ and } x_2 \leq 1.1) = 0.6962\end{split}$

### Compute the non central multivariate student’s t cdf¶

/*
** Upper limits of integration
** x1 ≤ -1 and x2 ≤ -1.1
** x1 ≤ 0 and x2 ≤ 0.1
** x1 ≤ 1 and x2 ≤ 1.1
*/
x = {  -1   -1.1,
0    0.1,
1    1.1 };

// Correlation matrix
corr = {  1 0.31,
0.31    1 };

// Define non-centrality vector, Kx1
nonc = {  1, -2.5 };

// Define degree of freedom
df  = 3;

// Define control structure
struct cdfmControl ctl;
ctl = cdfmControlCreate();

/*
** Calculate cumulative probability of
** each pair of upper limits
*/
{p, err, retcode}  = cdfMvte(ctl, x, corr, nonc, df);


After the above code, p should equal:

0.03571301
0.15854358
0.46919524


which means with non-central vector, the multivariate student’s t cdf are:

$\begin{split}P(x_1 \leq -1 \text{ and } x_2 \leq -1.1) = 0.0357\\ P(x_1 \leq +0 \text{ and } x_2 \leq +0.1) = 0.1585\\ P(x_1 \leq 1 \text{ and } x_2 \leq 1.1) = 0.4692\end{split}$

## References¶

1. 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, 63:361-378, 1999.
2. Genz, A., ‘’Numerical computation of multivariate normal probabilities,’’ Journal of Computational and Graphical Statistics, 1:141-149, 1992.