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.