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BUILT-IN MATHEMATICA SYMBOL
ExponentialFamily
ExponentialFamily
is an option for GeneralizedLinearModelFit that specifies the exponential family for the model.
Details
- 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->"QuasiLikelihood", 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->{"QuasiLikelihood", opts} 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.
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