Evaluation of Tweedie Exponential Family Models.

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.