Variance function

Variance Function new article content ...

Not to be confused with Variance.

In statistics, the variance function is a function relating the variance of a random quantity to the conditional mean of the random quantity. The variance function is a main ingredient in the generalized linear model framework and plays roles in Non-parametric regression and Functional data analysis as well. Not to be confused with the variance of a function, in parametric modelling, variance functions explicitly describe the relationship between the variance and the conditional mean of a random variable. For many well known distributions, the variance function represents the complete variance of a random variable under that distribution, but in fact, these are just special cases.

Intuition

Overview

Types

Generalized Linear Model

Derivation

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 of the form,

${\displaystyle \operatorname {f} (y,\theta ,\phi )=exp\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)}$ 

with loglikelihood,

${\displaystyle \operatorname {l} (\theta ,y,\phi )=\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)}$ 

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)

Failed to parse (syntax error): {\displaystyle \operatorname{E}_\theta\left[log(f_\theta(y)) }

Derivation

Examples

Normal

Binomial

Poisson

Gamma

Application

Maximum Likelihood Estimation

Quasi Likelihood

Non-Parametric Regression Analysis

See Also

References

External links

• example.com

Category:Generalized Linear Models Category:Quasi Likelihood Category:Non Parametric Regression