There is an interesting result, saying that if $Z_1, Z_2$ are standard normal random variables with a correlation $\rho\in (-1,1)$, then the product $Z=Z_1Z_2$ has a density function explicitly given by
$$ f_Z(z)=\frac{1}{\pi\sqrt{1-\rho^2}}\exp\left[\frac{\rho z}{1-\rho^2}\right]K_0\left(\frac{|z|}{1-\rho^2}\right), $$ where $K_0$ is the "modified Bessel function of the second kind of order zero" (Nadarajaha and Pogány (doi:10.1016/j.crma.2015.10.019))
My problem: I am looking for a function $g$ such that there exist a function $h$ such that $g(Z)=h(\rho)+\varepsilon$ for some random variable $\varepsilon$ which doesn't depend on $\rho$. So in a sense "dividing" the distribution on a "parameter" part and on a "random" part. Problem is that working with the modified Bessel function seems quite hard as it doesn't have a nice closed form.
On some examples, I will show what I am looking for more precisely if $Z$ would be in some other forms:
- If $Z\sim N(\rho, 1)$ then $g=identity$ and $Z=\rho + \varepsilon$ where $\varepsilon\sim N(0, 1)$.
- If $Z\sim Exp(\rho)$ then $g=-\log$ gives $-\log(Z) = \log(\rho) + \varepsilon$ where $\varepsilon\sim Gumbel(0,1)$.
I am looking for something like this but in our case when $Z$ having the distribution above.
Alternative question in terms of copulas: Doing the same for $Z=Z_1Z_2$ where $Z_1, Z_2$ are two correlated uniformly distributed r.v. on $[0,1]$. We know that the dependence structure between $Z_1, Z_2$ can be expressed in terms of copula, and there are many parametric forms of copulas. Now, pick some copula that you like (the more common one the better, and not some degenerative one) and assume $Z_1, Z_2$ follow it.
What I am looking for is such a case when we can find functions $g,h$ such that $g(Z_1, Z_2) = h(copula\, parameters) + \varepsilon$. In some vague words, can we "divide" it on a parameter PLUS a random part? (Here, I assume $g$ is a function of $Z_1, Z_2$, which is more allows more possibilities than just the product $Z_1Z_2$).
