I have to do a linear model with some variables $\mathbb X$ but I found out that the outcome $Y$ follows a very strange law, which is the Rayleigh distribution. Since the distribution belongs to the exponential family I could use a generalized linear model to find the estimations of the $\beta s$, but I do not know if it is better to use as output:
$Y$ without knowing how a linear model works with this type of variable?
$Y^2$ exploiting the fact that has $\chi^2(2)$ distribution (property of the Rayleigh) and use $Y^2$ as output of the linear model although it would assume very large values?
$\log(Y)$ which performs pretty good with the shapiro test (p-value=0.3) but is clearly not Gaussian?