Variance function: Difference between revisions

Revision as of 21:05, 27 February 2014

Revision as of 21:10, 27 February 2014

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The [[Generalized Linear Model]], '''GLM''', is a method of regression analysis that extends to any member of the [[exponential family]]. It is particularly useful when the response variable is categorical, binary or subject to a constraint (e.g. only positive responses make sense). See the page on [[generalized linear models]] for more information.

Any random variable in the exponential family has a probability density function and likelihood function of the form,

:<math>\operatorname{f}(y,\theta,\phi) = exp\left(\frac{y\theta - b(\theta)}{\phi} - c(y,\phi)\right)

</math>

andwith loglikelihood,

:<math>\operatorname{l}(\theta,y,\phi) = \left(\frac{y\theta - b(\theta)}{\phi} - c(y,\phi)\right)

</math>

We use the [['''Bartlett's Identities]] - insert reference'' to find the general '''variance function'''.

The first Bartlett results gives us that under suitable conditions ( '''insert references''')

:<math>\operatorname{E}_\theta\left[log(f_\theta(y))

</math>

We use the [[Bartlett Identities]] to find the general '''variance function'''.

In the '''GLM''' framework:

====Derivation====