For the gamma distribution, is the estimated dispersion parameter that is displayed in the output. The parameter
is also sometimes called the gamma index parameter.
For the negative binomial distribution, k is the estimated dispersion parameter that is displayed in the output.
The Tweedie model is a generalized linear model from the exponential family. The Tweedie distribution is characterized by
three parameters: the mean parameter , the dispersion
, and the power p. The variance of the distribution is
. For values of p in the range
, a Tweedie random variable can be represented as a Poisson sum of gamma distributed random variables. That is,
where N has a Poisson distribution
that has mean and the
have independent, identical gamma distributions
, each of which has an expected value
and an index parameter
.
In this case, Y has a discrete mass at 0, , and the probability density of Y
is represented by an infinite series for
. The HPGENSELECT procedure restricts the power parameter to satisfy
for numerical stability in model fitting. The Tweedie distribution does not have a general closed form representation for
all values of p. It can be characterized in terms of the distribution mean parameter
, dispersion parameter
, and power parameter p. For more information about the Tweedie distribution, see Frees (2010).
The distribution mean and variance are given by:
For the zero-inflated negative binomial distribution, k is the estimated dispersion parameter that is displayed in the output.