Tweedie Distributions - The Tweedie Convergence Theorem

The Tweedie Convergence Theorem

The Tweedie exponential dispersion models are fundamental in statistical theory consequent to their roles as foci of convergence for a wide range of statistical processes. Jørgensen et al proved a theorem that specifies the asymptotic behaviour of variance functions known as the Tweedie convergence theorem". This theorem, in technical terms, is stated thus: The unit variance function is regular of order p at zero (or infinity) provided that V(μ)~c₀μp for μ as it approaches zero (or infinity) for all real values of p and c₀ >0. Then for a unit variance function regular of order p at either zero or infinity and for

,

for any, and we have

as or, respectively, where the convergence is through values of c such that cμ is in the domain of θ and cp-2/σ2 is in the domain of λ. The model must be infinitely divisible as c2-p approaches infinity.

In nontechnical terms this theorem implies that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the domain of attraction of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.