Definitions

# Tweedie distributions

In probability and statistics, the Tweedie distributions are a family of probability distributions which include continuous distributions such as the normal and gamma, the purely discrete scaled Poisson distribution, and the class of mixed compound Poisson-Gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions belong to the exponential dispersion model family of distributions, a generalization of the exponential family, which are the response distributions for generalized linear models.

Tweedie distributions have a mean $mu$ and a variance $phi mu^p$, where $phi>0$ is a dispersion parameter, and $p$, called the index parameter, (uniquely) determines the distribution in the Tweedie family. Special cases include:

Tweedie distributions exist for all real values of $p$ except for

The Tweedie distributions were so named by Bent Jørgensen after M.C.K. Tweedie, a medical statistician at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1984.

The index parameter $p$ defines the type of distribution:

• For $p<0$, the data $y$ are supported on the whole real line (but, interestingly, $mu>0$). Applications for these distribution are unknown.
• For $p=0$ (the normal distribution), the data $y$ and the mean $mu$ are supported on the whole real line.
• For
• For $p=1$, the distribution exist on the non-negative integers
• For
• For $p>2$, the data $y$ are supported on the non-negative reals, and $mu>0$. These distribution are like the gamma distribution (which corresponds to $p=2$), but are progressively more right-skewed as $p$ gets larger.

## Applications

Applications of Tweedie distributions (apart from the four special cases identified) include:

• actuarial studies
• assay analysis
• survival analysis
• ecology
• analysis of alcohol consumption in British teenagers
• medical applications
• meteorology and climatology
• fisheries