cdfMvn2e#

Purpose#

Computes the multivariate normal cumulative distribution function with error management over the range \([a,b]\).

Format#

{ y, err, retcode } = cdfMvn2e(ctl, l_lim, u_lim, 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.

  • l_lim (NxK matrix) – lower limits.

  • u_lim (NxK matrix) – 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 inputs, u_lim and l_lim, have rows. \(Pr(X ≥ l\_lim \text{ and } X ≤ u\_lim|corr, nonc)\).

  • 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

    corr not positive semi-definite.

    missing

    corr not properly defined.

Examples#

Uncorrelated variables#

// Lower limits of integration for K dimensional multivariate distribution
a = {-1e4 -1e4};

// Upper limits of integration for K dimensional multivariate distribution
b = {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 from -1e4 to  0
*/
{ p, err, retcode } = cdfMvn2e(ctl, a, b, corr, nonc);

After the above code, both p equal to 0.25.

\[\Phi = P(-10000 \leq X_1 \leq 0 \text{ and } -10000 \leq X_2 \leq 0) \approx 0.25.\]

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

/*
** Limits of integration
** -5 ≤ x1 ≤ -1 and -8 ≤ x2 ≤ -1.1
** -10 ≤ x1 ≤ 0 and -10 ≤ x2 ≤ 0.1
** 0 ≤ x1 ≤ 1 and 0 ≤ x2 ≤ 1.1
*/
a = {  -5  -8,
      -20 -10,
        0   0 };

b = {  -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 limits
*/
{ p, err, retcode }  = cdfMvn2e(ctl, a, b, corr, nonc);

After the above code, p should equal:

0.04074118
0.31981965
0.13700266

which means that:

\[\begin{split}P(-5 \leq x_1 \leq -1 \text{ and } -8 \leq x_2 \leq -1.1) = 0.0407\\ P(-20 \leq x_1 \leq 0 \text{ and } -10 \leq x_2 \leq 0.1) = 0.3198\\ P(0 \leq x_1 \leq 1 \text{ and } 0 \leq x_2 \leq 1.1) = 0.1370\end{split}\]

Compute the non central multivariate normal cdf#

/*
** Limits of integration
** -5 ≤ x1 ≤ -1 and -8 ≤ x2 ≤ -1.1
** -10 ≤ x1 ≤ 0 and -10 ≤ x2 ≤ 0.1
** 0 ≤ x1 ≤ 1 and 0 ≤ x2 ≤ 1.1
*/
a = { -5  -8,
     -20 -10,
       0   0 };

b = {  -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 control structure
struct cdfmControl ctl;
ctl = cdfmControlCreate();

/*
** Calculate cumulative probability of
** each pair of upper limits
*/
{ p, err, retcode } = cdfMvn2e(ctl, a, b, corr, nonc);

After the above code, p should equal:

0.02246034
0.15854761
0.00094761

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

\[\begin{split}P(-5 \leq x_1 \leq -1 \text{ and } -8 \leq x_2 \leq -1.1) = 0.0225\\ P(-20 \leq x_1 \leq 0 \text{ and } -10 \leq x_2 \leq 0.1) = 0.1585\\ P(0 \leq x_1 \leq 1 \text{ and } 0 \leq x_2 \leq 1.1) = 0.0009\end{split}\]

Remarks#

  • cdfMvn2e() evaluates the following non-central MVN integral, where \(1\leqslant i \leqslant N\) 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.

See also

Functions cdfMvne(), cdfMvnce(), cdfMvt2e()

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