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 subfamiliy of (reproductive) exponential dispersion models (ED), with a special mean-variance relationship. A random variable Y is Tweedie distributed Twp(μ, σ2), if Y ∼ E D ( μ , σ 2 ) {displaystyle Ysim mathrm {ED} (mu ,sigma ^{2})} with mean μ = E ⁡ ( Y ) {displaystyle mu =operatorname {E} (Y)} , positive dispersion parameter σ 2 {displaystyle sigma ^{2}} and where p ∈ R {displaystyle pin mathbf {R} } 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 disperson model and cumulant function where we used α = p − 2 p − 1 {displaystyle alpha ={frac {p-2}{p-1}}} , or equivalently p = α − 2 α − 1 {displaystyle p={frac {alpha -2}{alpha -1}}} . 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. Note that all stable distributions mean actually generated by stable distributions.