Many of the probability distributions that the HPGENSELECT procedure fits are members of an exponential family of distributions, which have probability distributions that are expressed as follows for some functions b and c that determine the specific distribution:
For fixed , this is a one-parameter exponential family of distributions. The response variable can be discrete or continuous, so represents either a probability mass function or a probability density function. A more useful parameterization of generalized linear models is by the mean and variance of the distribution:
In generalized linear models, the mean of the response distribution is related to linear regression parameters through a link function,
for the ith observation, where is a fixed known vector of explanatory variables and is a vector of regression parameters. The HPGENSELECT procedure parameterizes models in terms of the regression parameters and either the dispersion parameter or a parameter that is related to , depending on the model. For exponential family models, the distribution variance is where is a variance function that depends only on .
The zero-inflated models and the multinomial models are not exponential family models, but they are closely related models that are useful and are included in the HPGENSELECT procedure.