I've been working on implementing a glm library in python for my students to use (for various reasons, the existing implementations are not sufficient). My reference for the material is Marlene Müller's paper.

Müller gives two formulas for estimating the dispersion in a fit glm. The first is in terms of the residual sum of squares:

$$ \hat a(\phi) = \frac{1}{n - p} \sum_i \frac{(y_i - \hat \mu_i)^2}{V(\hat \mu_i)} $$

And the second is in terms of the scaled deviance:

$$ \hat a(\phi) = \frac{D(y, \hat \mu)}{n - p} $$

At the moment, I've used the second formula, but not for any very good reason.

So my question is: are there reasons to prefer one of these estimators over the other? I would assume that they are both asymptotically correct, but is one better than the other in finite samples in any way?



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