< List of probability distributions <* Generalized Gamma Distribution*

The **generalized gamma distribution (GG)** was introduced by Stacy in 1962 [2] 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 [3].

## Properties of the generalized gamma distribution

The probability density function (PDF) for non-negative *x* from a generalized gamma distribution is [2]:

Where

*d*> 0 and*p*> 0 are shape parameters,*a*> 0 is a scale parameter,- Γ is the gamma function.

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 [4]

**f(x) = cx ^{δα−1}e^{−(xβ)δ}; x, α, β, δ > 0.**

The cumulative distribution function (CDF) is:

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:

If… | The generalized gamma distribution becomes the… |

d = p | Weibull distribution |

p = 1 | Gamma distribution |

p = d = 1 | Exponential distribution |

p = 2 and d = 2m | Nakagami distribution |

p = 2 and d = 1 | Half-normal distribution |

## Advantages and disadvantages

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 [5].

## History of the generalized gamma distribution

The GG distribution first appeared when in the early 20th century when L. Amoroso [6] and R. d’Addario [7] 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 [1] presenting the concept of the generalized gamma distribution and its fundamental characteristics. This publication marked the first comprehensive discussion of this distribution.

## References

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

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

[3] Mead et al. A Generalization of Generalized Gamma Distributions. Retrieved August 5, 2021 from: https://pjsor.com/pjsor/article/download/1692/635/

[4]Norman L. Johnson, Samuel Kotz, N. Balakrishnan, 1994, Continuous Univariate Distributions, Second edition, Vol. 1 Wiley Series in Probability and Mathematical Statistics.

[5] Matheson, M., Muñoz, A. & Cox, C. Describing the Flexibility of the Generalized Gamma and Related Distributions. *J Stat Distrib App* **4**, 15 (2017). https://doi.org/10.1186/s40488-017-0072-5

[6] Amoroso, L. Ricerche intorno alla curva dei redditi. *Annali di Matematica* **2**, 123–159 (1925). https://doi.org/10.1007/BF02409935

[7] D’ Addario, R. (1932). Intorno alla curva dei redditi di amoroso. Annali *di* mate matica pura ed applicata. serie IV t. II, 1924-25