Generalized gamma distribution
The generalized gamma distribution is a continuous probability distribution with three parameters. It is a generalization of the two-parameter gamma distribution. Since many distributions commonly used for parametric models in survival analysis (such as the Weibull distribution and the log-normal distribution) are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data.
where denotes the gamma function.
where denotes the lower incomplete gamma function.
Alternative parameterisations of this distribution are sometimes used; for example with the substitution α = d/p. In addition, a shift parameter can be added, so the domain of x starts at some value other than zero. If the restrictions on the signs of a, d and p are also lifted (but α = d/p remains positive), this gives a distribution called the Amoroso distribution, after the Italian mathematician and economist Luigi Amoroso who described it in 1925.
If X has a generalized gamma distribution as above, then
If and are the probability density functions of two generalized gamma distributions, then their Kullback-Leibler divergence is given by
- Box-Steffensmeier, Janet M.; Jones, Bradford S. (2004) Event History Modeling: A Guide for Social Scientists. Cambridge University Press. ISBN 0-521-54673-7 (pp. 41-43)
- Stacy, E.W. (1962). "A Generalization of the Gamma Distribution." Annals of Mathematical Statistics 33(3): 1187-1192. JSTOR 2237889
- Johnson, N.L.; Kotz, S; Balakrishnan, N. (1994) Continuous Univariate Distributions, Volume 1, 2nd Edition. Wiley. ISBN 0-471-58495-9 (Section 17.8.7)
- Gavin E. Crooks (2010), The Amoroso Distribution, Technical Note, Lawrence Berkeley National Laboratory.
- C. Bauckhage (2014), Computing the Kullback-Leibler Divergence between two Generalized Gamma Distributions, arXiv:1401.6853.