# Generalized Gamma Distribution

< List of probability distributions < Generalized Gamma Distribution

The generalized gamma distribution (GG) was introduced by Stacy in 1962  and encompasses various sub-models, including the exponential, gamma, Nakagami and Weibull distributions, among others.

The GG distribution is well-suited for modeling a diverse range of hazard rate functions, such as increasing, decreasing, bathtub, and arc-shaped. It has found applications in numerous research fields, including engineering, hydrology, and survival analysis .

## Properties of the generalized gamma distribution

The probability density function (PDF) for non-negative x from a generalized gamma distribution is :

Where

When d is between 0 and 1 the shape of the GG PDF is skewed right. The right skew becomes more pronounced as d → 0.

Other forms of the PDF do appear in the literature and depend on the different parameters. For example, one other possible form is 

f(x) = cxδα−1e−(xβ)δ; x, α, β, δ > 0.

where

• γ( · ) is the lower incomplete gamma function
• P( · , · ) is the regularized lower incomplete gamma function.

A number of familiar probability distributions can be obtained as special cases of the PDF by making certain choices for parameters:

The GG distribution is readily available in most popular statistical packages and is broadly applicable and flexible. However, there are many kinds of hazards, even among the four basic shapes, which it can’t accommodate .

## History of the generalized gamma distribution

The GG distribution first appeared when in the early 20th century when L. Amoroso  and R. d’Addario  used a form of the GG to analyze the distribution of economic income.

However, there wasn’t much interest in the GG until the mid-20th century when various properties, applications, and related distributions emerged. In 1962, Stacy published a paper  presenting the concept of the generalized gamma distribution and its fundamental characteristics. This publication marked the first comprehensive discussion of this distribution.

## References

 Fuzzyrandom, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

 Stacy, E.W. (1962). “A Generalization of the Gamma Distribution.” Annals of Mathematical Statistics 33(3): 1187-1192.