Gamma distribution

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The gamma distribution is a two-parameter, continuous distribution often used in statistical modeling because this flexible distribution can be used to model a wide variety of data.

The three-parameter gamma distribution is a generalization of the 2-parameter gamma distribution with an additional threshold parameter.

Gamma distribution properties

Formally, y is distributed as a gamma variable with shape parameter α > 0 and rate parameter β > 0 if it has the probability density function (PDF)

Where Γ is the gamma function, defined by the integral

If α = 1, the PDF is the exponential distribution; If β = 1, then this is the standard gamma distribution.

• Mean: E(X) = αβ
• Moment generating function (MGF): MX (t) = 1 /(1 − βt)^α.
• Variance: var(X) = αβ²

An alternate parameterization of the gamma distribution uses (k, θ), where k = α and θ = 1/β, has the PDF:

f(y; α, β) = (β^α) / Γ(α) * (y^{α – 1)} * e^(-βy)

The two forms are equivalent; which you use is a matter of choice.

What is the gamma distribution used for?

The gamma distribution can be used to model the same types of phenomena as the exponential distribution, including failure times, service times, and wait times. The most frequent use case is to model the time between independent events that occur at a constant average rate. Using this distribution, you can specify the number of events, such as modeling the time until the 3rd of 4th accident occurs. In this context, the gamma distribution models failure times in reliability analysis.

As a visualization of what the PDF means, α is the number of events in the queue (although α can be any positive number — not just integers), and β is the mean waiting time until the first event. If α remains unchanged but β increases, wait times have increased so the graph shifts to the right. Similarly, if β stays the same but α increases, the graph will also shift to the right. As α approaches infinity, the gamma closely matches the normal distribution.

The gamma distribution is often used in Bayesian inference. For example, y could be modeled as a mean measure of a count variable, which is the number of times an event occurs. You could use a gamma prior for the λ parameter of a Poisson distribution; That’s because the gamma distribution is only defined for λ≥ 0. The gamma distribution could also be used as a prior for a decision parameter — a quantity we want to estimate or make a decision about — because of its conjugate properties for precision modeling [2].

Related distributions

• The gamma distribution is a special case of the Tweedie distribution (when p = 2).
• For integer degrees of freedom, the Wishart distribution is the multivariate counterpart of the gamma distribution.
• The inverse gamma distribution, mainly used in Bayesian statistics, has the same distribution as the reciprocal of a gamma distribution.
• The Erlang distribution and chi-squared distribution are special cases of the gamma. The Erlang distribution happens when α is any positive integer.

Gamma distribution vs. normal distribution

The gamma and normal distribution are both members of the exponential family of distributions [4]. One difference between the two is how their parameters are defined: the normal distribution’s parameters are called the mean and standard deviation, while the gamma’s parameters are called shape and scale. A three parameter gamma distribution has an additional parameter called a threshold parameter, which determines the theoretical limits for the distribution.

The major difference is the shape of the distribution: the normal distribution is always symmetric around a single bell-shaped hump for the entire range of real numbers while the gamma may or may not have a hump, is not symmetric, and is only defined for positive real numbers [5].

References

[1] MarkSweep and Cburnettderivative work: Autopilot, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons

[2] Artemiou, A. (2009). Chapter 4 — Lecture 4 The Gamma Dist. and its Relatives. Retrieved November 3, 2017 from: http://www.math.mtu.edu/~aartemio/Courses/Stat318/Lectures/Chapter4/Chapter4_Lecture4.pdf.

[3] Gamma vs exponential graph retrieved 1/1/2023 from: http://homepages.cae.wisc.edu/~ie642/content/Techniques/Gamma/gamma_distribution.html

[4] Chapter 8: The exponential family: Basics.

[5] Getting Started with Gamma Regression. Retrieved August 18, 2023 from: https://library.virginia.edu/data/articles/getting-started-with-gamma-regression