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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?

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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.

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  • $\begingroup$ Excellent point, I haven't considered this before. Is there a workaround to the GLM methodology in cases where the distribution is not of the exponential family? $\endgroup$
    – jon
    Mar 23, 2014 at 14:32
  • $\begingroup$ Generalized Estimating Equations (GEE). You only need the first two moments to use them. $\endgroup$ Mar 23, 2014 at 19:33

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