I am interesting in how to differentiate multivariate normal cdf respect to diagonal elements of covariance matrix (that is, I am interested only in variances).
Problem similar to mine has been discussed in https://stats.stackexchange.com/questions/47624/how-do-you-calculate-the-derivative-of-the-multivariate-normal-cdf-with-respect but here is a formula only for correlations (it works correct for covariations as well without need in standartization) while I am interested in dispersions.
So the formula from the topic is (it works correct even for covariations)
$\frac{\partial F(x;0;\Sigma)}{\partial\rho_{i,j}}=f(x_{i},x_{j};0;\Sigma_{i,j})F(x-\{x_{i},x_{j}\}|x_{i},x_{j})$
But I need
$\frac{\partial F(x;0;\Sigma)}{\partial Var_{i}}$
I have been trying to solve the task setting $i=j$ to the topic formula but did not scucceed.
Will be very grateful for help!
The answer I have recieved seems to be a little bit complicated but may be it can't be derived simplier. I will show it on example with 3 dimensions but it could be easily generalized to multivariate case.
$\frac{\partial F((x_{1},x_{2},x_{3});0;\Sigma)}{\partial Var(x_{1})}=\frac{1}{2}\frac{\partial^2\int_{-\inf}^{x_{1}}\int_{-\inf}^{x_{2}}\int_{-\inf}^{x_{3}}f(v_{1},v_{2},v_{3})dv_{1}dv_{2}dv_{3}}{\partial x_{1}^2}=\frac{1}{2}\frac{\partial\int_{-\inf}^{x_{2}}\int_{-\inf}^{x_{3}}f(v_{2},v_{3}|x_{1})dv_{2}dv_{3}f(x_{1})}{\partial x_{1}}=\frac{1}{2}(F(x_{2},x_{3}|x_{1})*\frac{\partial f(x_{1})}{\partial x_{1}}+f(x_{1})*\frac{\partial F(x_{2},x_{3}|x_{1})}{\partial x_{1}})$
Any answers with simplier formula are welcome!
P.S. You may check that formula is correct using finite difference method