Great! The doc on the link of shape says “if predicted” which we don’t do in this situation. Since this question has come up multiple times now, I will adjust the doc to make this clearer.
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With regard to other families, it is certaintly implementable for some of them, if they have a closed form expression of the sum distribution, such as for exponential and gamma. However, until someone actually requires it, I don’t want to invest the time figuring the details out for each family.
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You may want to look at this paper:
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Hello, just wondering, is it not the same as to talk something similar to quasi-Poisson when the rate is scaled with N? When it is not scaled, then it is Neg. Binomial.