Exponential Dispersion Models

An exponential dispersion model adds a second parameter, the dispersion index, to a natural exponential family (here and ). Let be a unit cumulant function and , with , a convolution semigroup of base measures (). The additive model is the exponential family with log-partition ,

with mean and variance . Its reproductive form () is with mean , dispersion , and , where the unit variance function is .

Reproductive property

If are independent with and , then . At a common mean the precisions add. Under addition the index set is a commutative monoid, a convex cone, acting by convolution, and the weighted average is its projectivisation.

Example, continuous index and the Tweedie class

The additive model extends to a continuous index, a Lévy convolution semigroup , iff is a Lévy exponent, equivalently the generating law is infinitely divisible. Among natural exponential families the reproducible ones are exactly the power-variance Tweedie models, with (Bar-Lev–Enis), obeying the scaling . The index flow’s generator is a genuine transport velocity iff the Lévy measure vanishes, giving Gaussian ; otherwise it is a jump process.