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Exponential dispersion model

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Exponential dispersion models are statistical models in which the probability distribution is of a special form. This class of models represents a generalisation of the exponential family of models which themselves play an important role in statistical theory because they have a special structure which enables deductions to be made about appropriate statistical inference.

Definition

Exponential dispersion models are a generalisation of the natural exponential family: these have a probability density function which, for a multivariate model, can be written as

f X ( x | θ ) = h ( x ) exp ⁡ ( θ ⊤ x − A ( θ ) ) ,

where the parameter θ has the same dimension as the observation variable x . The generalisation includes an extra scalar "index parameter", λ , and has density function of the form

f X ( x | λ , θ ) = h ( λ , x ) exp ⁡ ( λ [ θ ⊤ x − A ( θ ) ] ) .

The terminology "dispersion parameter" is used for σ 2 = λ − 1 , while θ is the "natural parameter" (also known as "canonical parameter").

References

Exponential dispersion model Wikipedia
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