Variance function: Difference between revisions

Revision as of 07:18, 1 March 2014

Revision as of 18:44, 4 March 2014

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=====Maximum Likelihood Estimation=====

=====Quasi Likelihood=====

Variance functions play a very important role in [[Quasi-likelihood]] estimation. [[Quasi-likelihood]] estimation is useful when there appears to be [[overdispersion]] in the data, or when [[overdispersion]] is likely. Overdispersion occurs when there is more variability in the data thenthan there should otherwise be expected according to the assumed distribution of the data. This can happen for many reasons, one common reason being that thethere datais high correlation between data points (grouped data). Because most features of '''GLMs''' only depend on the first two moments of the distribution, rather than then entire distribution, the Quasi-likelihood can be developed by just specifiyingspecifying a link function and a variance function. That is we need to specify

: - Link Function: <math>E[y] = \mu = g^{-1}(\eta)</math>

: - Variance Function: <math>V(\mu)\text{, where the }Var_\theta(y) = \sigma^2V(\mu)</math>

With a specified variance function and link function we can develop, as alternatives to the log-[[likelihood function]], the [[score function]], and the [[Fisher information]], a '''[[Quasi-likelihood]]''', a '''Quasi-score''', and the '''Quasi-Information'''. This allows for full inference of <math>\beta</math>.

:'''Quasi-Likelihood'''

With a specified variance function and link function we can develop, as an alternative the [[score function]], a '''Quasi-score'''

Though called a [[Quasi-likelihood]], this is in fact a quasi-'''log'''-likelihood.

:'''Quasi-Score'''

:<math>U = \frac{y-\mu}{\sigma^2V(\mu)}</math>

The first two Bartlett equations are satisfied for the Quasi-Score, namely

:<math> E[U] = 0 </math> and

:<math> cov(U) + E[\frac{\partial U}{\partial \mu}] = 0.</math>

In addition, the the quasi-score is linear in '''y'''.

===Non-Parametric Regression Analysis===