Chernoff's distribution

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

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

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

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

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

References

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