Cfrgtkky

Say we have some function of bivariate standard normal random variables $f(x_1,x_2)$ which we approximate with first 2 terms of its Taylor series,

$$z = f(x_1,x_2) approx f(0,0) + f_1(0,0)x_1 + f_2(0,0)x_2 + 0.5f_{1,1}(0,0)x_1^2 + 0.5f_{2,2}(0,0)x_2^2 + f_{1,2}(0,0)x_1x_2$$

The unconditional expectation of this approximation can be computed using Isserlis' theorem (which is more relevant for when higher terms are retained in the Taylor series and the function depend on more than 2 normals) as

$$E[z] approx f(0,0) + 0.5f_{1,1}(0,0) + 0.5f_{2,2}(0,0) + f_{1,2}(0,0)rho .$$

But how does one compute a conditional expectation

$$E[z | z in A] space approx space ?.$$

Can this be done without having to work-out the explicit from of the pdf for $z$? Or perhaps there is already a known closed form expression for the distribution of a linear combination of monomials of multivariate normal rvs?

Generally, the function will depend on a multivariate normal and higher order of the Taylor series may be kept.

Add 1

What might be possible to do is to workout the first $m$ moments (using Isserlis' theorem to calculate expectations of various monomials), then convert them into cumulants, and then use Edgeworth series approximation (or something like that) to construct approximate "distribution function" for $z$, and then calculate the conditional expectation. It seems rather convoluted however. Also, since conditioning could be on event in the tail, it might require quite a few terms in the approximation.