Likelihood expression for functions of Gumbel random variables in stan

Modeling
1

[Please include Stan program and accompanying data if possible]
Hello, I wanna ask how to write the likelihood expression for functions of Gumbel random variables.

Specifically, my problem is: y is my observation and
y =\integrate alphax / (thetax+beta^2+gamma) by beta and gamma
where x is my data, alpha and theta are the parameters I wanna estimate. they follow standard normal distribution. beta and gamma are unobserved parameters and follow iid standard Gumbel distribution. I’d like to ask the likelihood expression for y in stan. Your answer is highly appreciated. thanks!

2

guys, I need your help, thanks!

3

Hi Alex, you might want to start by writing up a simpler Stan model for your data so you have a handle on how values get plumbed through a density calculation in Stan. There is no E operator in Stan as you write so beta and gamma might need to be unobserved parameters. I think if you get a simpler program put together you’ll be able to ask a more specific question.

4

thanks for your reply! here x is my data, alpha and theta are the parameters that I wanna estimate, beta and gamma are unobserved parameters that I wanna take their expectation

6

I’ve revised my question. I hope it would help

7

Even if the beta and the gamma aren’t in this problem, you’re probably going to have a tough time. If we just fix beta == 0 and gamma == 0, then:

y = alpha * x / (theta * x) = alpha / theta

So it’s gonna be tough just from observations of y to figure out what alpha and theta are doing (especially if x is big). Where’d this problem come from?