cdfMvne#

Purpose#

Computes multivariate Normal cumulative distribution function with error management.

Format#

{ y, err, retcode } = cdfMvne(ctl, x, corr, nonc)#
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

    error tolerance.

  • x (1xK vector or NxK matrix) – Upper limits at which to evaluate the lower tail of the multivariate normal cumulative distribution function. x must have K columns–one for each variable. If x has more than one row, each row will be treated as a separate set of upper limits.

  • corr (KxK matrix) – correlation matrix.

  • nonc (Kx1 vector) – non-centrality vector.

Returns:

  • p (Nx1 vector) – Each element in p is the cumulative distribution function of the multivariate normal distribution for each corresponding columns in x. p will have as many elements as the input, x, has columns.

  • 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 distribution
x = { 0  0 };

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

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

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

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

/*
** Calculate joint probability of two
** variables with zero correlation,
** both, being ≤ 0
*/
p2 = cdfn(0) .* cdfn(0);

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

\[\Phi = P(-\infty < X_1 \leq 0 \text{ and } - \infty < X_2 \leq 0) \approx 0.25.\]

Compute the multivariate normal 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 control structure
struct cdfmControl ctl;
ctl = cdfmControlCreate();

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

After the above code, p should equal:

0.040741382
0.31981965
0.74642007

which means that:

\[\begin{split}P(x_1 \leq -1 \text{ and } x_2 \leq -1.1) = 0.0407\\ P(x_1 \leq +0 \text{ and } x_2 \leq +0.1) = 0.3198\\ P(x_1 \leq 1 \text{ and } x_2 \leq 1.1) = 0.7464\end{split}\]

Compute the non central multivariate normal 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 parameter for each variable
nonc  = { 1, -2.5 };

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

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

After the above code, p should equal:

0.02246034
0.15854761
0.49998320

which means with non-central vector, the multivariate normal cdf are:

\[\begin{split}P(x_1 \leq -1 \text{ and } x_2 \leq -1.1) = 0.0225\\ P(x_1 \leq +0 \text{ and } x_2 \leq +0.1) = 0.1585\\ P(x_1 \leq 1 \text{ and } x_2 \leq 1.1) = 0.5000\end{split}\]

Remarks#

  • cdfMvne evaluates the MVN integral, where \(1\leqslant i \leqslant N\) For the non-central MVN we have where \(z\) denotes \(K\) -dimensional multivariate normal distribution, denotes the \(K \times 1\) non-centrality vector with \(-\infty<\:\ \delta_k <\:\ \infty\) .

  • The correlation matrix \(R\) is defined by \(\Sigma = DRD\), where \(D\) denotes the diagonal matrix which has the square roots of the diagonal entries for covariance matrix \(\Sigma\) on its diagonal.

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 cdfMvne(), cdfMvn2e(), cdfMvte()

On this page