# 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.