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:
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:
domain for responses
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.