Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models. 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 (reproductive) Tweedie distributions are defined as subfamily of (reproductive) exponential dispersion models (ED), with a special mean-variance relationship. A random variable Y is Tweedie distributed Twp(μ, σ2), if with mean , positive dispersion parameter and where is called Tweedie power parameter. The probability distribution Pθ,σ2 on the measurable sets A, is given by for some σ-finite measure νλ. This representation uses the canonical parameter θ of an exponential dispersion model and cumulant function where we used , or equivalently . The models just described are in the reproductive form. An exponential dispersion model has always a dual: the additive form. If Y is reproductive, then with is in the additive form ED*(θ,λ), for Tweedie Twp(μ, λ). Additive models have the property that the distribution of the sum of independent random variables, for which Zi ~ ED(θ,λi) with fixed θ and various λ are members of the family of distributions with the same θ, A second class of exponential dispersion models exists designated by the random variable where σ2 = 1/λ, known as reproductive exponential dispersion models. They have the property that for n independent random variables Yi ~ ED(μ,σ2/wi), with weighting factors wi and a weighted average of the variables gives, For reproductive models the weighted average of independent random variables with fixed μ and σ2 and various values for wi is a member of the family of distributions with same μ and σ2.