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

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

where the parameter $\mathit{\theta }$ 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

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

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

References

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