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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 | | Hello from Great Britain. I'm glad to came across you. My first name is Carley. <br>I live in a small town called Scalby in nothern Great Britain.<br>I was also born in Scalby 32 years ago. Married in February 2007. I'm working at the the office.<br><br>Feel free to visit my web site ... [http://flashpixie.com/profile/mibaggett wordpress backup] |
| 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]].
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| ==Definition==
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| 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
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| :<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/>
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| :<math> f_X(\mathbf{x}|\lambda,\boldsymbol{\theta}) = h(\lambda,\mathbf{x}) \exp (\lambda [\boldsymbol\theta^\top \mathbf{x} - A(\boldsymbol\theta)] ) \,\! .</math>
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| 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").
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| ==References==
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| {{reflist}}
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| [[Category:Statistical models]]
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| [[Category:Statistical theory]]
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Latest revision as of 14:10, 5 May 2014
Hello from Great Britain. I'm glad to came across you. My first name is Carley.
I live in a small town called Scalby in nothern Great Britain.
I was also born in Scalby 32 years ago. Married in February 2007. I'm working at the the office.
Feel free to visit my web site ... wordpress backup