<List of probability distributions < *Johnson’s S _{B} distribution*

## What is Johnson’s S_{B} distribution?

**Johnson’s S _{B} distribution** (or “special bounded” distribution) is a four-parameter distribution used to model different types of bounded data in a wide variety of fields, including finance, quality control, and reliability engineering.

Johnson’s S_{B} distribution is part of *Johnson’s system*, which contains three other distributions: the log-normal distribution, the normal distribution, and Johnson’s S_{U} distribution, which models unbounded systems. Each member of the system can transform to a normal distribution by using elementary functions. In other words, every member of Johnson’s system is a transformation of a normal distribution [1]. The transformation for an S_{B} to normal distribution is invertible (i.e. reversible), which means that you can use normal distribution properties to understand S_{B} distribution properties and generate random S_{B} distribution variates from random normal variates.

The distribution is named after Norman L. Johnson, who introduced it in 1949.

## Johnson’s S_{B} distribution properties

The Johnson family of distributions is defined by

, with *z* = *γ* + *δf*(*Y*)

. *Y *= (*X* −* xi*)/*lambda*

The Johnson S_{B} distribution arises when

, where *f*(*Y*)=*ln*[*Y*/(1−*Y*)]`0 < `

. This is a bounded distribution because the range of Y is *Y *< 1`(0,1)(0,1)`

[2].

Johnson [3] showed that bounded random variables with upper and lower limits could transform to an approximately normal distribution with the following transformation:

Z has a probability density function (PDF) of

Which leads to the following equation for Johnson’s S_{B} distribution:

or

This is a continuous distribution defined on bounded range ε ≤ *x* ≤ ε + λ; the distribution can be either symmetric or asymmetric.

## Uses for Johnson’s S_{B} distribution

Olsson [4] developed a powerful computer program capable of fitting Johnson’s S_{B} System and S_{U} Systems with Nelder-Mead simplex subroutines. The Nelder-Mead simplex algorithm is a derivative-free algorithm, which means that it does not require the calculation of the derivatives of the objective function (the function that is being minimized). This means that the Nelder-Mead simplex algorithm is robust and can be used to fit functions with noisy data. Olsson generated maximum likelihood estimates for the four necessary parameters in this procedure, demonstrating that it is an invaluable tool when dealing with grouped data sets.

This application has also been used prominently within forestry literature thanks to Hafley and Schreuder [5], who first introduced its capabilities over three decades ago. Parresol [6] furthered understanding by showcasing how recovery of vital distributional parameters can be gained from the system itself – inspiring more reliable analysis overall.

Other uses of Johnson’s *S _{B}* distribution include fitting human exposure data in epidemiology [7].

## References

[1] The Johnson SB distribution – The DO Loop – SAS Blogs. Retrieved August 6, 2023 from: https://blogs.sas.com/content/iml/2020/01/20/johnson-sb-distribution.html

[2] Z. A. Karian and E. J. Dudewicz (2011) Handbook of Fitting Statistical Distributions with R, Chapman & Hall.

[3] Johnson, N.L. 1949. Systems of frequency curves generated by methods of translation. Biometrika. 36: 149-176.

[4] Olsson, D. (1979). Fitting Johnson’s *S _{B}* and

*S*Systems of Curves using the Method of Maximum Likelihood, Journal of Quality Technology, 11, 211-217.

_{U}[5] Hafley, W.L.; Schreuder, H.T. 1977. Statistical distributions for fitting diameter and height data in even-aged stands. Canadian Journal of Forest Research. 7: 481-487.

[6] Parresol, B. (2003). Recovering Parameters of Johnson’s *S _{B}* distribution. USDA.

[7] Flynn, M. (2005). Fitting Human Exposure Data with the Johnson SB Distribution. Journal of Exposure Science & Environmental Epidemiology 16(4)