Supposing a vector x follows normal distribution. I want to calculate the expectation of the "variance Of the covariance matrix" (not variance-covariance matrix) in a vector form, meaning E[vech(x x')vech(x x')'], given that we know E[x x'] = S and the expectation of x is zero vector. This is a peace of a fisher information matrix.
Anyone can give me some suggestion on how to get that? Thanks for your help.
This requires computation of fourth order statistics. Use the property that if $x_1$, $x_2$, $x_3$, $x_4$ are normal random variables then
$$ E[x_1 x_2 x_3 x_4] = E[x_1 x_2] E[x_3 x_4] + E[x_1 x_3]E[x_2 x_4] + E[x_1 x_4]E[x_2 x_3] $$
This will allow you to write $E[vech(x x')vech(x x')']$ in terms of the elements of $S$. Some work will be required to rewrite the result in matrix form.