Subsections:

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.