tweedie
Evaluation of Tweedie Exponential Family Models
Maximum likelihood computations for Tweedie families, including the series expansion (Dunn and Smyth, 2005;
tweedie 
The tweedie package allows likelihood computations for Tweedie distributions.
Apart from special cases (the normal, Poisson, gamma, inverse Gaussian distributions), Tweedie distributions do not have closed-form density functions or distribution functions. This package uses fast numerical algorithms (infinite oscillation integrals; infinite series) to evaluate the Tweedie density functions and distribution functions.
Installation
You can install the development version of tweedie from GitHub with:
# install.packages("pak")
pak::pak("PeterKDunn/tweedie")
Tweedie distributions
Tweedie distributions are exponential dispersion models, with a mean $\mu$ and a variance $\phi \mu^\xi$, for some dispersion parameter $\phi > 0$ and a power index $\xi$ (sometimes called $p$) that uniquely defines the distribution within the Tweedie family (for all real values of $\xi$ not between 0 and 1).
Special cases of the Tweedie distributions are:
- the normal distribution, with $\xi = 0$ (i.e., the variance is $\phi$ and not related to the mean);
- the Poisson distribution, with $\xi = 1$ and $\phi = 1$ (i.e., the variance is the same as the mean);
- the gamma distribution, with $\xi = 2$; and
- the inverse Gaussian distribution, with $\xi = 3$.
For all other values of $\xi$, the probability functions and distribution functions have no closed forms.
For $\xi < 1$, applications are limited (non-existent so far?), but have support on the entire real line and $\mu > 0$.
For $1 < \xi < 2$, Tweedie distributions can be represented as a Poisson sum of gamma distributions. These distributions are continuous for $Y > 0$ but have a discrete mass at $Y = 0$.
For $\xi \ge 2$, the distributions have support on the positive reals.
The vignette contains examples.