I am fitting a Weibull distribution to some data in Stan. I am trying to reproduce some published values of parameters from a paper. However I am running into problems because I believe the normalizing constant does not match. The paper gives the pdf equation as follows:
$p(x|\mu,\nu) = \frac{1}{K} (\frac{x}{\nu})^{\mu - 1} exp(-(\frac{x}{\nu})^{\mu})$
However the Weibull pdf in Stan is:
$p(x|\mu,\nu) = \frac{\mu}{\nu} (\frac{x}{\nu})^{\mu - 1} exp(-(\frac{x}{\nu})^{\mu})$
When I fit the distribution to the same data as the paper, I get different fitted values for the shape and scale parameters ($\mu$ and $\nu$) than the ones in the paper, but the paper gives no indication of how to find the normalizing constant K. Is there a way to determine the correct value for the constant so that I can get the correct values of shape and scale parameters?
Here is the (very simple) Stan model I fit:
data {
int<lower=0> N;
vector<lower=0>[N] x;
}
parameters {
// Weibull density
real<lower=0> mu;
real<lower=0> nu;
}
model {
// Priors: Weibull density
mu ~ lognormal(1, 1);
nu ~ lognormal(1, 1);
// Likelihood: Weibull density
x ~ weibull(mu, nu);
}