Support for zero-augmented Poisson distributions

[zeileis]

Alex @alexpghayes , as already mentioned in the discussion about the Poisson distribution vignette, I have now put together distributions3 support for the two standard zero-augmented Poisson distributions:

• Zero-inflated Poisson (ZIP), combining a standard Poisson distribution with parameter lambda with excess zeros with probability pi. Thus, there is a mixed source of zeros: Poisson vs. excess.
• Hurdle Poisson, combining zeros with a probability 1 - pi with a left-truncated (at zero) Poisson distribution with parameter lambda. Thus, pi is the hurdle-crossing probability at zero and the only source of zeros.

For both distributions I have added d/p/q/r functions for zipois and hpois, respectively, following the style of the corresponding pois functions. Then I have added ZIPoisson() and HurdlePoisson() constructors along with the corresponding methods following the style of Poisson(). This infrastructure will greatly facilitate working with probabilistic regression models based on these distributions.

[alexpghayes]

Wonderful, thank you!

[zeileis]

Perfect, thanks! 🎉

The next step would be to provide the same for the zero-augmented negative binomial distributions. This is pretty much straightforward, I just have to do the algebra to get the skewness and kurtosis (if I can) which might be tricky.

And then only the mean+size parameterization is left for the negative binomial in order to support the standard count regression distributions. Do you have any thoughts/preferences regarding this? I think it would be relatively straightforward to allow the same NegativeBinomial() creator function with either p + size or with mu + size because in the background the same d/p/q/r functions for the nbinomial are called. Should I have a stab at this?

[alexpghayes]

Sounds good to me!