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 and
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:
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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.