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I've fitted a non-exponential family GLM regression model with the response distributed as a t-distribution with $\nu$ degrees of freedom, scale $\theta$ and mean $\mu = X\beta$.

We estimate $\beta,\nu,$ and $\theta$.

I get $\theta = 0.6$, $\nu = 4.9$, and also my $\beta$ values.

I want to check my distributional assumption in a qqplot. In order to do so, I need to compute normalized quantile residuals. Any idea how?

I tried the following, and got good results, but I would like you to check my approach:

  1. First, define $F_i(y; \mu, \theta)$ as the cumulative distribution function of a scaled t-distribution with degrees of freedom 4.9 and scale 0.6 and mean parameter equal to the $i$th fitted value from my model.
  2. Then, take the inverse standard normal cumulative distributional function on each $F_i$.

Is that the way to do it?

Orbit
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    Could you explain why in step (2) you are proposing to apply a standard *Normal* distribution as a reference for residuals that you assume have a *Student t* distribution? – whuber Jun 05 '17 at 17:12
  • I believe my steps (1) and (2) correspond to the inner resp. outer function in the formula in page 3 here: http://www.statsci.org/smyth/pubs/residual.pdf, the idea being that this transformation, given that the model is correct, should produce normally distributed residuals which can then be plotted in a standard qqplot. – Orbit Jun 05 '17 at 18:33
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    My question is why you would care about normally distributed residuals at all when your model explicitly rejects that as a behavior for them. Since your reference distribution is a Student t distribution, *compare your residuals to the Student t*, not to the Normal! – whuber Jun 05 '17 at 18:51
  • Hmm ... Could you point to where you think I assume the residuals follow a $t$-distribution? The model is basically a GLM, except with a non-exponential family distribution. And I am under the impression that nothing is directly assumed of residuals of GLMs, unlike in a linear model. – Orbit Jun 05 '17 at 20:29
  • https://stats.stackexchange.com/questions/92394/checking-residuals-for-normality-in-generalised-linear-models – Orbit Jun 05 '17 at 20:34
  • As far as I can see, the point of the answer there is to use a Normal p-p plot to show that residuals are *not* normal (and then to show a case in which they happen to be so). That doesn't seem to be your purpose. Your purpose ought to be to compare your residuals to the model assumptions, which are decidedly *not* about normality. – whuber Jun 05 '17 at 21:02
  • 1) The very purpose of this question is to **transform** my residuals so that they *are* about normality. 2) You claim that I can just compare my residuals to the $t$-distribution instead. Implicitly you're saying my model assumes that my residuals actually are $t$-distributed. Why should they be? They might not be normally distributed true, but that's not the same as them being $t$-distributed just because that's the distribution $y$ follows. – Orbit Jun 05 '17 at 22:00
  • It is true that the residual distribution will depend on the fitting method. But since your model is that the errors are t-distributed, then a Student t is your reference distribution. It's hard to see the point of trying to turn the residuals into a normal distribution. What's the problem with stopping at step 1 and comparing those values to a uniform distribution? – whuber Jun 05 '17 at 22:56

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