Define sigma in mvncdf function in Matlab

So for 2 rho coefficients we have:

   p = mvncdf([X1, X2], [0, 0], [1 rho; rho 1])

But for 3 rho coefficients I am not sure how to define the sigma: I have X1, X2, and X3, with mean zero [0, 0, 0] and rho coefficients

rho_112, rho_113 and rho_123

how do I define it in the function:

   p = mvncdf([X1, X2, X3], [0, 0, 0], [1 rho; rho 1])

Also very curious why we sometimes need a minus before the rho

Rho in this case is: How to compute lower tail probability for the Bivariate Normal Distribution

Solution

You should be aware that the density function for a multivariate normal distribution is: $f_X(x_1, x_2, \ldots, x_k) = \frac{\exp\left(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)}{\sqrt{(2\pi)^k|\Sigma|}}$

Hence, when $\mu =0$ , it would be:

$f_X(x_1, x_2, \ldots, x_k) = \frac{\exp\left(-\frac{1}{2}x^T\Sigma^{-1}x\right)}{\sqrt{(2\pi)^k|\Sigma|}}$

On other hand, as $\rho_{ij}$ is the correlation coefficient between $x_i$ and $x_j$ , we can write the covarience matrix like the following:

As $\sigma_i =1$ for all $i$ , we can write the covariance matrix like the following:

Therefore, for a three dimensional distribution, you can have:

p = mvncdf([X1, X2, X3], [0, 0, 0], [1 rho12 roh13; rho21 1 rho23; rho31 rho32 1])

Notice that $\rho_{ijk}$ is not useful here. It is a correlation between three variables, but the covariance matrix just needs mutual covariances.