In probability and statistics, the class of exponential dispersion models (EDM), also called exponential dispersion family (EDF), is a set of probability distributions that represents a generalisation of the natural exponential family.
[1][2][3] 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.
There are two versions to formulate an exponential dispersion model.
In the univariate case, a real-valued random variable
belongs to the additive exponential dispersion model with canonical parameter
and index parameter
λ
( θ , λ )
, if its probability density function can be written as The distribution of the transformed random variable
is called reproductive exponential dispersion model,
θ = (
The terminology dispersion model stems from interpreting
as dispersion parameter.
For fixed parameter
σ
is a natural exponential family.
In the multivariate case, the n-dimensional random variable
has a probability density function of the following form[1] where the parameter
θ
The cumulant-generating function of
θ = (
Mean and variance of
are given by with unit variance function
are called reproductive.
The probability density function of an
can also be expressed in terms of the unit deviance
as where the unit deviance takes the special form
or in terms of the unit variance function as
{\frac {y-t}{V(t)}}\,dt}
Many 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.