# How do you get quantile normalized residuals for a t-distribution fit?

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?

• 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 5 '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: 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 5 '17 at 18:33
• 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 5 '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 5 '17 at 20:29
• stats.stackexchange.com/questions/92394/… – Orbit Jun 5 '17 at 20:34