cdfMvn#

Purpose#

Computes multivariate Normal cumulative distribution function.

Format#

p = cdfMvn(x, corr)#

Parameters:

x (Kx1 vector or KxN matrix) – Values at which to evaluate the multivariate normal cumulative distribution function. If x has more than one column, each column will be treated as a separate set of upper limits.
corr (KxK matrix) – correlation matrix.

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.

Examples#

Uncorrelated variables#

// Upper limits of integration
x = { 0, 0 };

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

/*
** Calculate cumulative probability of
** both variables being ≤ 0
*/
p = cdfmvn(x, corr);

/*
** Calculate joint probablity 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.

Example 2#

// Upper limits of integration
x = { -0.5, 1 };

// Correlation matrix
corr = {   1  0.26,
        0.26     1 };

/*
** Calculate cumulative probability of
** the first variable being ≤ -0.5
** and the second variable being ≤ 1
*/
p = cdfmvn(x, corr);

After the above code, p should equal: 0.28025. It means :

\[\Phi = P(-\infty < X_1 \leq -0.5, - \infty < X_2 \leq 1) \approx 0.28025\]

when the correlation between two variables is 0.26.

Compute the cdf at 3 separate pairs of points#

/*
** Upper limits of integration
** x1 ≤ -1 and x2 ≤ -1.1
** x1 ≤  0 and x2 ≤  0.1
** x1 ≤  1 and x2 ≤  1.1
*/
x = {  -1   0    1,
     -1.1 0.1  1.1 };

// Correlation matrix
corr = {   1 0.31,
        0.31    1 };

/*
** Calculate cumulative probability of
** each pair of upper limits
*/
p = cdfmvn(x, corr);

After the above code, p should equal:

0.040741382   0.31981965   0.74642007

which means that:

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

Remarks#

cdfMvn() evaluates the MVN integral with i-th row of \(x\) (upper limits), where \(1\leqslant i \leqslant N\)
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.
cdfMvne() is more accurate and faster. Note that cdfMvne() takes a row vector of upper limits whereas cdfMvn() takes a column vector of limits.