# cdfMvnce¶

## Purpose¶

Computes the complement (upper tail) of the multivariate Normal cumulative distribution function with error management.

## Format¶

{ y, err, retcode } = cdfMvnce(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 scalar, error tolerance.

• x (NxK matrix) – Lower limits at which to evaluate the complement 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) – $$Pr(X ≥ x|corr, nonc)$$. The probability in the upper tail of the multivariate normal cumulative distribution function for each corresponding set of limits 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¶

// Lower 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 parameter for each variable
nonc  = { 0, 0};

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

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

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


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

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

### Compute the upper tail of multivariate normal cdf at 3 separate pairs of lower limits¶

/*
** Lower 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  = { 0, 0 };

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

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


After the above code, p should equal:

0.74642007
0.27999181
0.04074138


which means that:

$\begin{split}P(x_1 \geq -1 \text{ and } x_2 \geq -1.1) = 0.7464\\ P(x_1 \geq +0 \text{ and } x_2 \geq +0.1) = 0.2800\\ P(x_1 \geq 1 \text{ and } x_2 \geq 1.1) = 0.0407\end{split}$

### Compute the upper tail of noncentral multivariate normal cdf¶

/* Lower 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
// vector, one for each variable
nonc  = { 1, -2.5 };

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

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


After the above code, p should equal:

0.08046686
0.00455354
0.00014231


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

$\begin{split}P(x_1 \geq -1 \text{ and } x_2 \geq -1.1) = 0.0805\\ P(x_1 \geq +0 \text{ and } x_2 \geq +0.1) = 0.0046\\ P(x_1 \geq 1 \text{ and } x_2 \geq 1.1) = 0.0001\end{split}$

## Remarks¶

• The cdfMvnce() evaluates the upper tail of MVN integral, where $$1\leqslant i \leqslant N$$ For the non-central MVN, we have where $$z$$ denotes $$K$$ -dimensional multivariate normal distribution, $$\delta$$ 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.