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| | '''Exponential dispersion models''' are [[statistical model]]s in which the probability distribution is of a special form.<ref>Marriott, P. (2005) "Local Mixtures and Exponential Dispersion |
| | Models" [http://www.stat.duke.edu/~paul/Paperspdf/dispersion.pdf pdf]</ref><ref name=J1987>Jørgensen, B. (1987). Exponential dispersion models (with discussion). [[Journal of the Royal Statistical Society]], Series B, 49 (2), 127–162.</ref> 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 |
| | :<math> f_X(\mathbf{x}|\boldsymbol{\theta}) = h(\mathbf{x}) \exp(\boldsymbol\theta^\top \mathbf{x} - A(\boldsymbol\theta)) \,\! ,</math> |
| | where the parameter <math>\boldsymbol\theta</math> has the same dimension as the observation variable <math>\mathbf{x}</math>. The generalisation includes an extra scalar "index parameter", <math>\lambda</math>, and has density function of the form<ref name=J1987/> |
| | :<math> f_X(\mathbf{x}|\lambda,\boldsymbol{\theta}) = h(\lambda,\mathbf{x}) \exp (\lambda [\boldsymbol\theta^\top \mathbf{x} - A(\boldsymbol\theta)] ) \,\! .</math> |
| | The terminology "dispersion parameter" is used for <math>\sigma^2=\lambda^{-1}</math>, while <math>\boldsymbol\theta</math> is the "natural parameter" (also known as "canonical parameter"). |
| | |
| | ==References== |
| | {{reflist}} |
| | |
| | [[Category:Statistical models]] |
| | [[Category:Statistical theory]] |
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
where the parameter 
has the same dimension as the observation variable 
. The generalisation includes an extra scalar "index parameter", 
, and has density function of the form[2]
![{\displaystyle f_{X}(\mathbf {x} |\lambda ,{\boldsymbol {\theta }})=h(\lambda ,\mathbf {x} )\exp(\lambda [{\boldsymbol {\theta }}^{\top }\mathbf {x} -A({\boldsymbol {\theta }})])\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16a21fbadb1aa3f2ecb8c29b43723951e665110c)
The terminology "dispersion parameter" is used for 
, while is the "natural parameter" (also known as "canonical parameter").
References
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