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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:
0≤y≤1
  • 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:
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Fit data to a simple linear regression model:
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Fit to a canonical gamma regression model:
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Fit to a canonical inverse Gaussian regression model:
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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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