# How do you calculate the derivative of the multivariate normal CDF with respect to a correlation coefficient?

How does one calculate the derivative of a multivariate normal CDF with respect to a given correlation coefficient?

I have started with the bivariate case but couldn't work it out.

• Do you really mean CDF? Or do you rather mean PDF? Jan 13, 2013 at 8:50
• No, I really mean CDF! Differentiating the PDF would be easy. I have tried this for the bivariate case, by taking the derivative inside the integral and then differentiating the PDF. But then the algebra gets really hard when I try to solve the integral. Jan 13, 2013 at 9:39
• just a remak in the case where you do not know this: the cdf does not exist in closed form Jan 13, 2013 at 10:02
• I'm not sure what you're trying to do or why (I'm not sure what it means to take a derivative "with respect to a given correlation coefficient")... can you explain? Jan 14, 2013 at 3:13
• The what is to calculate $dF(x)/d\rho_{ij}$, the why is to be able to learn the parameters of a product of Gaussian copulae by gradient ascent. Jan 14, 2013 at 4:01

$\frac{\partial}{\partial\rho_{ij}}f(x;0,\Sigma)=\frac{\partial^2}{\partial x_i\partial x_j}f(x;0,\Sigma)$ ("A reduction formula for normal multivariate integrals", Plackett 1954).
$\frac{\partial}{\partial\rho_{ij}}F(x;0,\Sigma)=f(x_i,x_j;0,\Sigma_{ij})F(x-\{x_i,x_j\}\ |\ x_i,x_j)$