This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 BUILT-IN MATHEMATICA SYMBOL

# ExponentialFamily

 ExponentialFamily is an option for GeneralizedLinearModelFit that specifies the exponential family for the model.
• ExponentialFamily specifies the assumed distribution for the independent observations modeled by .
• The density function for an exponential family can be written in the form for functions , , , , and , random variable , canonical parameter , and dispersion parameter .
• Possible parametric distributions include: , , , , .
• The observed responses are restricted to the domains of parametric distributions as follows:
 "Binomial" "Gamma" "Gaussian" "InverseGaussian" "Poisson"
• The setting ExponentialFamily, defines a quasi-likelihood function, used for a maximum likelihood fit.
• The log quasi-likelihood function for the response and prediction is given by , where is the dispersion parameter and is the variance function. The dispersion parameter is estimated from input data and can be controlled through the option DispersionEstimatorFunction.
• The setting ExponentialFamily allows the following quasi-likelihood suboptions to be specified:
 "ResponseDomain" Function[y,y>0] domain for responses "VarianceFunction" Function[,1] variance as function of mean
• The parametric distributions can be emulated with quasi-likelihood structures by using the following and suboption settings:
• variants of and families can be used to model overdispersed () or underdispersed () data, different from the theoretical dispersion ().
• Common variance functions, response domains, and uses include:
 power models, actuarial science, meteorology, etc. probability models, binomial related, etc. counting models, Poisson related, etc.
Fit data to a simple linear regression model:
Fit to a canonical gamma regression model:
Fit to a canonical inverse Gaussian regression model:
Fit data to a simple linear regression model:
 Out[2]=
Fit to a canonical gamma regression model:
 Out[3]=
Fit to a canonical inverse Gaussian regression model:
 Out[4]=
 Scope   (2)
Use the family for logit models of probabilities:
Use for log-linear models of count data:
The default model matches LogitModelFit:
Fit a model and the analog:
The models differ from named analogs by a constant in the :
Fitted parameters agree:
Results based on differences of log-likelihoods agree:
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