Generalized Gamma Distribution

Basic Concepts

The generalized gamma distribution is a three-parameter version of the gamma distribution. It adds a second shape parameter δ. A number of distributions besides the gamma distribution are special cases of this distribution.

The probability density function (pdf) of the generalized gamma distribution is

Generalized gamma distribution pdf

In this case, we use the notation x ~ G(α, β, δ).

The cumulative distribution function (cdf) is given by

Generalized gamma distribution cdf

where γ(α, z) is the lower incomplete gamma function (see Gamma Function Advanced).

If x has a gamma distribution, then x ~ G(α, β, 1), and so, the pdf is

Gamma distribution pdf

as expected.

Properties

Property 1: If k > 0 and x ~ G(α, β, δ), then

kxG(α, kβ, δ)          xkG(α, βk, δ/k)

Proof: Since x ~ G(α, β, δ), for the second proposition

f(x^k)

The proof of the first proposition is similar.

Property 2: If y = (x/β)δ and x ~ G(α, β, δ), then y has a gamma distribution x ~ G(α, 1, 1)

Proof: This follows from Property 1.

Thus, if F(x) is the cdf of G(α, β, δ) and H is the cdf of y = (x/β)δ, then F(x) = H(y). It also follows that

Property 3: If F(x) is the cdf of G(α, β, δ) and H is the cdf of y = (x/β)δ, then

F-1(p) = β ⋅ (H-1(p))1/δ

Related Distributions

Property 4: If x ~ G(1, β, δ) then x has a Weibull distribution (with alpha = β and beta = δ).

Proof: The pdf is

Weibull pdf

Property 5: If x ~ G(1, β, 1) then x has an exponential distribution (with lambda = 1/β).

Proof: This follows since an exponential distribution is equivalent to a gamma distribution with α = 1 and β = 1/λ.

It also follows from Property 4 since an exponential distribution is equivalent to a Weibull distribution (with alpha = β and beta = 1). Thus

Weibull to Exponential distribution

Property 6: If x ~ G(1/2, β, 2) then x has a half-normal distribution

Half-normal pdf part 1

Here we are using the fact that Γ(1/2) = √π (see Gamma Function)  Letting σ = β/√2, and so β = σ √π, we have

Half-normal distribution part 2

Moments

Mean

Mode

Variance

Skewness

Kurtosis

where

Kurtosis A

Kurtosis B

Kurtosis C

Kurtosis D

Kurtosis E

The following is a generalization of the formula for the mean:

kth Moment

Worksheet Functions

Real Statistics Functions: Starting with Rel 8.10, the Real Statistics Resource Pack will provide the following functions for the generalized gamma distribution:

GAMMA_DIST(x, α, β, δ, cum) = the pdf of the generalized gamma function f(x)  when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

GAMMA_INV(p, α, β, δ) = x such that GAMMA_DIST(x, α, β, δ, TRUE) = p. Thus, GAMMA_INV is the inverse of the CDF of the generalized gamma distribution.

Examples Workbook

Click here to download the Excel workbook with some examples that accompany this webpage.

References

Nematrian (2023) The generalised gamma distribution
http://www.nematrian.com/GeneralisedGammaDistribution

Jimenez Nava, V. H. (2011) Gamma and generalized gamma distributions
https://scholarworks.utep.edu/open_etd/2321/

Wikipedia (2023) Generalized gamma distribution
https://en.wikipedia.org/wiki/Generalized_gamma_distribution

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