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2

It should be intuitively clear that the time from the first success to the second success is independent of the time from the start to the first success, and similarly for all non-overlapping combinations of success start and ends. This enables us to break the problem up into four independent sub-problems as follows: \mathbb{E}[\text{time to 4th success}]...

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Are you referring to regression models for count data? If yes, I think your best bet is to use the R packages gamlss and gamlss.dist. The gamlss package contains a gamlss() function which you can use to fit your regression models. The gamlss.dist package contains two functions which implement the distributions you need: NBI() implements the Negative ...

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Dispersion values will never be exactly 1, due to random variation in the data. Both tests don't seem to indicate overdispersion, although I would note that you don't really know for the function that you use, as it doesn't produce p-values. I believe performance::check_overdispersion is based on the same idea as your parametric test, but has p-values, so ...

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