Exponential dispersion model

Exponential dispersion models are statistical models in which the probability distribution is of a special form.^[1][2] 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

$f_X(\mathbf{x}|\boldsymbol{\theta}) = h(\mathbf{x}) \exp(\boldsymbol\theta^\top \mathbf{x} - A(\boldsymbol\theta)) \,\! ,$

where the parameter $\boldsymbol\theta$ has the same dimension as the observation variable $\mathbf{x}$. The generalisation includes an extra scalar "index parameter", $\lambda$, and has density function of the form^[2]

$f_X(\mathbf{x}|\lambda,\boldsymbol{\theta}) = h(\lambda,\mathbf{x}) \exp (\lambda [\boldsymbol\theta^\top \mathbf{x} - A(\boldsymbol\theta)] ) \,\! .$

The terminology "dispersion parameter" is used for $\sigma^2=\lambda^{-1}$, while $\boldsymbol\theta$ is the "natural parameter" (also known as "canonical parameter").

References

1. Marriott, P. (2005) "Local Mixtures and Exponential Dispersion Models" pdf
2. Jørgensen, B. (1987). Exponential dispersion models (with discussion). Journal of the Royal Statistical Society, Series B, 49 (2), 127–162.