Generalized gamma distribution

The generalized gamma distribution is a continuous probability distribution with two shape parameters. It is a generalization of the gamma distribution which has one shape parameter. Since many distributions commonly used for parametric models in survival analysis are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data. Another example is the half-normal distribution.

Characteristics

The generalized gamma distribution has two shape parameters, and, and a scale parameter,. For non-negative x from a generalized gamma distribution, the probability density function is
where denotes the gamma function.
The cumulative distribution function is
where denotes the lower incomplete gamma function,
and denotes the regularized lower incomplete gamma function.
The quantile function can be found by noting that where is the cumulative distribution function of the gamma distribution with parameters and. The quantile function is then given by inverting using known relations about inverse of composite functions, yielding:
with being the quantile function for a gamma distribution with.

Related distributions

If then the generalized gamma distribution becomes the Weibull distribution.
If the generalised gamma becomes the gamma distribution.
If then it becomes the exponential distribution.
If then it becomes the Rayleigh distribution.
If and then it becomes the Nakagami distribution.
If and then it becomes a half-normal distribution.

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, this gives a distribution called the Amoroso distribution, after the Italian mathematician and economist Luigi Amoroso who described it in 1925.

Moments

If X has a generalized gamma distribution as above, then

Properties

Denote GG as the generalized gamma distribution of parameters a, d, p.
Then, given and two positive real numbers, if, then
and

Kullback–Leibler divergence

If and are the probability density functions of two generalized gamma distributions, then their Kullback–Leibler divergence is given by
where is the digamma function.

Software implementation

In the R programming language, there are a few packages that include functions for fitting and generating generalized gamma distributions. The package in R allows for fitting and generating many different distribution families including . Other options in R, implemented in the package flexsurv, include the function dgengamma, with parameterization:,,, and in the package ggamma with parametrisation:,,.
In the python programming language, in the SciPy package, with parametrisation:,, and scale of 1.