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.

See also

Functions cdfMvte(), cdfMvt2e(), cdfMvnce()