Exponential dispersion model

In probability and statistics, the class of exponential dispersion models (EDM) is a set of probability distributions that represents a generalisation of the natural exponential family.
Exponential dispersion models play an important role in statistical theory, in particular in generalized linear models because they have a special structure which enables deductions to be made about appropriate statistical inference. 
Definition.
Univariate case.
There are two versions to formulate an exponential dispersion model.
Additive exponential dispersion model.
In the univariate case, a real-valued random variable formula_1 belongs to the additive exponential dispersion model with canonical parameter formula_2 and index parameter formula_3, formula_4, if its probability density function can be written as
Reproductive exponential dispersion model.
The distribution of the transformed random variable formula_6 is called reproductive exponential dispersion model, formula_7, and is given by
with formula_9 and formula_10, implying formula_11.
The terminology "dispersion model" stems from interpreting formula_12 as "dispersion parameter". For fixed parameter formula_12, the formula_14 is a natural exponential family.
Multivariate case.
In the multivariate case, the n-dimensional random variable formula_15 has a probability density function of the following form
where the parameter formula_17 has the same dimension as formula_15.
Properties.
Cumulant-generating function.
The cumulant-generating function of formula_19 is given by
with formula_11
Mean and variance.
Mean and variance of formula_19 are given by
with unit variance function formula_24.
Reproductive.
If formula_25 are i.i.d. with formula_26, i.e. same mean formula_27 and different weights formula_28, the weighted mean is again an formula_29 with
with formula_31. Therefore formula_32 are called "reproductive".
Unit deviance.
The probability density function of an formula_14 can also be expressed in terms of the unit deviance formula_34 as
where the unit deviance takes the special form formula_36 or in terms of the unit variance function as formula_37.
Examples.
A lot of very common probability distributions belong to the class of EDMs, among them are: normal distribution, Binomial distribution, Poisson distribution, Negative binomial distribution, Gamma distribution, Inverse Gaussian distribution, and Tweedie distribution.