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(−∞<X1≤0 and −∞<X2≤0)≈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:

P(x1≤−1 and x2≤−1.1)=0.0675P(x1≤+0 and x2≤+0.1)=0.3182P(x1≤1 and x2≤1.1)=0.6962

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:

P(x1≤−1 and x2≤−1.1)=0.0357P(x1≤+0 and x2≤+0.1)=0.1585P(x1≤1 and x2≤1.1)=0.4692

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()

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