# cdfMvt2e¶

## Purpose¶

Computes multivariate Student’s t cumulative distribution function with error management over $$[a, b]$$.

## Format¶

{ y, err, retcode } = cdfMvt2e(ctl, l_lim, u_lim, 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, relative error tolerance.
• l_lim (NxK matrix) – lower limits. K is the dimension of multivariate Student’s t distribution. N is the number of MVT cdf integrals.
• u_lim (NxK matrix) – upper limits.
• corr (KxK matrix) – correlation matrix.
• nonc (Kx1 vector) – noncentralities.
• df (scalar) – degrees of freedom.
Returns:
• p (Nx1 vector) – Each element in p is the cumulative distribution function of the multivariate Student’s t distribution for the corresponding columns in l_lim and u_lim. p will have as many elements as the input, l_lim and u_lim, have rows.
• 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¶

// Lower limits of integration for K dimensional multivariate distribution
l_lim = { -1e4 -1e4 };

// Upper limits of integration for K dimensional multivariate distribution
u_lim = { 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 from -1e4 to 0
*/
{ p, err, retcode } = cdfMvt2e(ctl, l_lim, u_lim, corr, nonc, df );


After the above code, both p 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¶

/*
** Limits of integration
** -5 ≤ x1 ≤ -1 and -8 ≤ x2 ≤ -1.1
** -20 ≤ x1 ≤ 0 and -10 ≤ x2 ≤ 0.1
**  0 ≤ x1 ≤ 1 and 0 ≤ x2 ≤ 1.1
*/
l_lim = { -5  -8,
-20 -10,
0   0 };

u_lim = { -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
** both variables being from -1e4 to 0
*/
{ p, err, retcode } = cdfMvt2e(ctl, l_lim, u_lim, corr, nonc, df );


After the above code, p should equal:

0.06226091
0.31743546
0.12010880


which means that:

$\begin{split}P(-5 \leq x_1 \leq -1 \text{ and } -8 \leq x_2 \leq -1.1) = 0.0623\\ P(-20 \leq x_1 \leq +0 \text{ and } -10 \leq x_2 \leq +0.1) = 0.3174\\ P(0 \leq x_1 \leq 1 \text{ and } 0 \leq x_2 \leq 1.1) = 0.1201\end{split}$

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

/*
** Limits of integration
** -5 ≤ x1 ≤ -1 and -8 ≤ x2 ≤ -1.1
** -20 ≤ x1 ≤ 0 and -10 ≤ x2 ≤ 0.1
**  0 ≤ x1 ≤ 1 and 0 ≤ x2 ≤ 1.1
*/
l_lim = { -5  -8,
-20 -10,
0   0 };

u_lim = { -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
** both variables being from -1e4 to 0
*/
{ p, err, retcode } = cdfMvt2e(ctl, l_lim, u_lim, corr, nonc, df );


After the above code, p should equal:

0.02810292
0.15190018
0.00092484


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

$\begin{split}P(-5 \leq x_1 \leq -1 \text{ and } -8 \leq x_2 \leq -1.1) = 0.0281\\ P(-20 \leq x_1 \leq +0 \text{ and } -10 \leq x_2 \leq +0.1) = 0.1519\\ P(0 \leq x_1 \leq 1 \text{ and } 0 \leq x_2 \leq 1.1) = 0.0009\end{split}$

## Source¶

cdfm.src

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.