# 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:

• $p=0$ is the normal distribution
• $p=1$ with $\phi =1$ is the Poisson distribution
• $p=2$ is the gamma distribution
• $p=3$ is the inverse Gaussian distribution.

Tweedie distributions exist for all real values of $p$ except for $0 . Apart from the four special cases identified above, their probability density function have no closed form. However, software is available that enables the accurate computation of the Tweedie densities (and probability distribution functions).

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 $0 , no distributions exist
• For $p=1$ , the distribution exist on the non-negative integers
• For $1 , the distribution is continuous on the positive reals, plus an added mass (exact zero) at $Y=0$ . For example, consider monthly rainfall. When no rain is recorded, an exact zero is recorded. If rain is recorded, a continuous amount results. These distributions are also called the Poisson-gamma distributions, since they can be represented as the Poisson sum of gamma distributions. They are therefore a type of compound Poisson distribution.
• 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:

• analysis of alcohol consumption in British teenagers 
• medical applications 
• meteorology and climatology 