Tweedie distributions

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal and gamma distributions, 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. For any random variable Y that obeys a Tweedie distribution, the variance var(Y) relates to the mean E(Y) by the power law,

where a and p are positive constants.

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

The Tweedie distributions include a number of familiar distributions as well as some unusual ones, each being specified by the domain of the index parameter. We have the

For 0 < p < 1 no Tweedie model exists.

Tweedie distributions are a special case of exponential dispersion models, a class of models used to describe error distributions for the generalized linear model. The term "exponential dispersion model" refers to the exponential form that these models take, evident from the canonical equation used to describe the distribution P_λ,θ of the random variable Z on the measurable sets A,

with the interrelated measures ν_λ. θ is the canonical parameter; the cumulant function is

...
Wikipedia