Exponential dispersion model

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.

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

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

...
Wikipedia