The Waring distribution is a generalization of the Simon-Yule distribution. To put that another way, the Simon-Yule is a special case of the Waring distribution when the parameters take on certain limiting values. The Waring has two parameters (α and β); The generalized form has an additional parameter ν.

The Waring is a theoretical distribution with limited (if any) applications to real-life data. Sichel (1992, as cited in Hahn & Keeble, 1998) 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.” It is sometimes linked to the Pareto distribution, because the tails show Pareto-like behavior (Arnold, 2015, p. 292).

## Disambiguation

The term “Waring Distribution” usually refers to a generalization of the Yule distribution, but many variants on the name exist, so it can get a little confusing. To add to the confusion, subtle changes in the generating mechanism for the Simon-Yule distribution (a.k.a. a specific form of the Waring) lead to various other distributions (e.g. the Poisson distribution). The generalized Waring distribution is sometimes called the *beta negative binomial distribution*.

When working with this distribution, make sure you understand the context. For example, in the Wolfram documentation, “*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.

## References

Applied Mathematics and Computation. 217 (21): 8560–8566.

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

Hazewinkel, M. (2001). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media.

King, M. (2017). Statistics: A Practical Approach for Process Control Engineers. John Wiley and Sons.

Wolfram language and documentation center. WaringYuleDistribution. Retrieved March 29, 2019 from: https://reference.wolfram.com/language/ref/WaringYuleDistribution.html

Yule, G. U. (1925). “A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S”. Philosophical Transactions of the Royal Society B. 213 (402–410): 21–87