Waring Distribution

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The Waring Distribution is a generalization of the Yule-Simon distribution, allowing for more flexibility and greater accuracy when modeling data. It is similar to the Pareto distribution because the tails show Pareto-like behavior.

About the Waring distribution

waring distribution
Waring distribution, Yule-Simon, and generalized Waring pdfs [1].

The Waring (or Beta-K) distribution has two parameters, α and β, which are analogous to the mean and variance of a normal distribution. The generalized form of the Waring distribution adds an additional parameter ν, which controls the shape of the distribution.

Applications of the Waring distribution

The Waring Distribution is a theoretical distribution with many applications in statistics, finance, engineering, computer science and economics. It is especially useful in regression analysis due to its ability to accurately model nonlinear relationships between variables. It can also be used in hypothesis testing since it allows for higher levels of accuracy than traditional parametric tests such as t-tests or F-tests.

The Waring distribution can be an invaluable tool for predicting future trends or patterns in data sets with high variability or nonlinearity. For instance, if you have a dataset with high variability that contains sales figures from different stores over time, you could use the Waring distribution to forecast future sales figures more accurately than would be possible with traditional parametric methods.

Other than a few use cases, the Waring isn’t suitable for modeling a wide swath real-life data. Sichel, as cited in [1], states that it has

“…linear tails in a logarithmic grid, and hence [is] unsuitable for representing the upper tails of most observed biometric size-frequency data.”


Many variants on the “Waring” name exist, so it can get a little confusing. For example, the generalized Waring distribution is sometimes called the beta negative binomial distribution.

In the Wolfram documentation [2], “Waring-Yule Distribution” refers to the Yule–Simon distribution:

  • WaringYuleDistribution[α] represents the Yule distribution with shape parameter α.
  • WaringYuleDistribution[α,β] represents the Waring with shape parameters α and β.


[1] Hahn, T. & Buckland, (1998). Historical Studies in Information Science. Information Today, Inc.

[2] Wolfram Research (2010), WaringYuleDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WaringYuleDistribution.html.

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