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I have a dataset with outcomes from a Vasicek distribution (see this pdf) and some covariates. Re-expressing the Vasicek's pdf into the exponential family form requires me to transform my data, i.e. instead of $y.\theta + c(\theta) + d(y)$, I have $a(y).\theta + c(\theta) + d(y)$, where $a(y) = \Phi^{-1}(y)$ is the inverse normal cdf function.
Will transforming my data by $\Phi^{-1}(y)$ and then fitting it to a GLM violate any assumptions?
No. However, it will make interpretation more difficult.
In a GLM we are trying to specify the mean-variance relationship (so that we can weight by the inverse of the variance). Normally, we let $E[Y]=\mu$ and ${\rm Var}[Y]=V(\mu)$. It looks as though you will have $E[a(Y)]=\mu$ and ${\rm Var}[a(Y)]=V(\mu)$. When you fit your model, you will have $\text{link}(E[a(Y)])=X\beta$ where $E[a(Y)]\neq a(E[Y])$. So, for example, if your link is the identity link, you could say that for two groups, one with an $x$ which is one unit higher, $E[a(Y)]$ is $\beta$ greater, but you cannot move the $a()$ and talk about $E[Y]$.
Also, looking at the paper you linked to, depending on what part of it you are using, I would be careful to ensure your $y$s are independent, if they are not, you need to do something more complicated.
