Tweedie Distribution
Y ~ Tw_p(μ, φ)
Exponential Dispersion / Compound Poisson-Gamma Family • Published July 22, 2026
The exponential dispersion family unifying five already-published shapes as exact special cases of a single mean-variance power law Var(Y)=φμ^p: the already-published Normal (p=0), Poisson (p=1), the already-published Gamma (p=2), and the already-published Inverse Gaussian (p=3), with the actuarially important 1<p<2 range giving a genuinely new compound-Poisson-Gamma shape with a point mass at zero plus a continuous density, verified via an infinite-series density formula matched exactly against a reference implementation, profile-likelihood parameter estimation, and simulation via the compound Poisson-Gamma construction.
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About This Reference Sheet
The exponential dispersion family unifying five already-published shapes as exact special cases of a single mean-variance power law Var(Y)=φμ^p: the already-published Normal (p=0), Poisson (p=1), the already-published Gamma (p=2), and the already-published Inverse Gaussian (p=3), with the actuarially important 1<p<2 range giving a genuinely new compound-Poisson-Gamma shape with a point mass at zero plus a continuous density, verified via an infinite-series density formula matched exactly against a reference implementation, profile-likelihood parameter estimation, and simulation via the compound Poisson-Gamma construction.
Support
y = 0 with positive probability, plus a continuous density on (0, ∞) -- for 1<p<2
Parameters
μ > 0 (mean), φ > 0 (dispersion), p ∉ (0,1) (power, fixed per model)