cdfTvn#

Purpose#

Computes the cumulative distribution function of the standardized trivariate Normal density (lower tail).

Format#

p = cdfTvn(x1, x2, x3, rho12, rho23, rho13)#

Parameters:

x1 (Nx1 vector) – upper limits of integration for variable 1
x2 (Nx1 vector) – upper limits of integration for variable 2
x3 (Nx1 vector) – upper limits of integration for variable 3
rho12 (scalar or Nx1 vector) – correlation coefficients between the two variables x1 and x2
rho23 (scalar or Nx1 vector) – correlation coefficients between the two variables x2 and x3
rho13 (scalar or Nx1 vector) – correlation coefficients between the two variables x1 and x3

Returns:

p (Nx1 vector) – result of the triple integral from \(-\infty\:\ to\:\ x_1\), \(-\infty\:\ to\:\ x_2\), and \(-\infty\:\ to\:\ x_3\) of the standardized trivariate Normal density.

Examples#

// Variables
x1 = 0.6;
x2 = 0.23;
x3 = 0.46;

//Correlations
rho12 = 0.2;
rho23 = 0.65;
rho13 = 0.78;

/*
** Compute the CDF
*/
p = cdfTvn(x1, x2, x3, rho12, rho23, rho13);
print "p =" p;

After the above code, X will equal:

p =  0.4373

Remarks#

Allowable ranges for the arguments are:

\[\begin{split}−\infty < x1 < \infty\\ −\infty \lt x2 \lt \infty\\ −\infty \lt x3 \lt \infty\\ −1 \lt rho12 \lt 1\\ −1 \lt rho23 \lt 1\\ −1 \lt rho13 \lt 1\\\end{split}\]

In addition, rho12, rho23 and rho13 must come from a legitimate positive definite matrix. A -1 is returned for those rows with invalid inputs.

A separate integral is computed for each row of the inputs.

To find the integral under a general trivariate density, with x1, x2, and x3 having nonzero means and any positive standard deviations, transform by subtracting the mean and dividing by the standard deviation. For example:

\[x1 = \frac{(x1 ⁢− meanc(x1))}{stdc(x1)}\]

The absolute error for cdfTvn() is approximately ±2.5e-8 for the entire range of arguments.

References#

Daley, D.J. ‘’Computation of Bi- and Tri-variate Normal Integral.’’ Appl. Statist. Vol. 23, No. 3, 1974, 435-38.
Steck, G.P. ‘’A Table for Computing Trivariate Normal Probabilities.’’ Ann. Math. Statist. Vol. 29, 780-800.