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 that 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 first referred to by that name by Bent Jørgensen in a 1987 paper, crediting Maurice Tweedie, a statistician and medical physicist at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1982 at the Indian Statistical Institute Golden Jubilee International Conference in Calcutta. In 1986, Shaul K. Bar-Lev and Peter Enis published a paper about the same topic in The Annals of Statistics.
Definitions 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 Y ~ ED(μ, σ2) with mean μ = E(Y), positive dispersion parameter σ2 and
Var ( Y ) = σ 2 μ p , {\displaystyle \operatorname {Var} (Y)=\sigma ^{2}\mu ^{p},}
where p ∈ R is called the Tweedie power parameter. The probability distribution Pθ,σ2 on the measurable sets A, is given by
P θ , σ 2 ( Y ∈ A ) = ∫ A exp ( θ ⋅ z − κ p ( θ ) σ 2 ) ⋅ ν λ ( d z ) , {\displaystyle P_{\theta ,\sigma ^{2}}(Y\in A)=\int _{A}\exp \left({\frac {\theta \cdot z-\kappa _{p}(\theta )}{\sigma ^{2}}}\right)\cdot \nu _{\lambda }\,(dz),}
for some σ-finite measure νλ. This representation uses the canonical parameter θ of an exponential dispersion model and cumulant function
κ p ( θ ) = { α − 1 α ( θ α − 1 ) α , for p ≠ 1 , 2 − log ( − θ ) , for p = 2 e θ , for p = 1 {\displaystyle \kappa _{p}(\theta )={\begin{cases}{\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha },&{\text{for }}p\neq 1,2\\-\log(-\theta ),&{\text{for }}p=2\\e^{\theta },&{\text{for }}p=1\end{cases}}}
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}}} .
Properties
Additive exponential dispersion models 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 Z = λY with λ = 1/σ2 is in the additive form ED*(θ, λ), for Tweedie Tw*p(μ, λ). Additive models have the property that the distribution of the sum of independent random variables,
Z + = Z 1 + ⋯ + Z n , {\displaystyle Z_{+}=Z_{1}+\cdots +Z_{n},}
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