< List of probability distributions

Probability distribution is a crucial element in statistics, which helps in understanding the behavior of random variables. In this context, Johnson SU distribution is one of the commonly used types of probability distribution. The Johnson SU distribution is widely applied in statistical modeling and statistical inference. The article explains what the Johnson SU distribution is, its properties, and how it is used.

## What is the Johnson SU distribution?

The Johnson SU distribution is an unbounded and continuous probability distribution; bounded distributions have constraints — such as height, time, or weight maximums and minimums — while unbounded distributions do not have these restrictions.

This distribution is typically associated with four parameters -γ, δ, λ, and ξ – which determine its shape, location, scale, and type. These four parameters provide a basis for understanding the behavior of the distribution. The presence of these four parameters in the probability distribution makes it flexible and suitable for fitting real-world data. This distribution is also used to transform random variables, making the normality assumption in statistical models plausible.

The probability density function (PDF) is

Johnson devised the Johnson distributions (SB (bounded), SL (lognormal) and SU) in the late 1940s to apply well-established normal distribution methods and theory to a wide range of phenomena via a series of simple transformations. He describes the SU distribution simply as [2]

Where λ and ε are shape parameters.

The Johnson SU Distribution is member of the *Johnson system*, a family of four probability distributions that includes

- Johnson SU distribution (U = “unbounded”),
- Johnson SB distribution (B = “bounded”),
- Lognormal distribution,
- Normal distribution.

One of the fundamental properties of the Johnson SU distribution is that it can take many shapes, ranging from a normal distribution to symmetric and asymmetric types, depending on its parameter setting. It also has a unique probability density function (PDF), which can closely fit the data with empirical values. In practice, the Johnson SU distribution has been applied in a wide range of research areas, including economics, engineering, and finance. The Johnson SU distribution can fit data that is leptokurtic and skewed [3]. That has made it useful in a variety of areas including modeling asset returns for portfolio management [4] and Value at Risk (VaR) modeling [5].

## Advantages of Johnson SU distribution

The Johnson SU distribution has many advantages over other distributions, such as the normal distribution. For instance, it offers the possibility of symmetrical or asymmetric distribution, explicitly fitting for both positive and negative tails of the data. This flexibility is crucial in statistical modeling to capture different types of variability present in observational data. Furthermore, it provides higher accuracy than other distribution types, making it easier to interpret the underlying data.

Another advantage of the Johnson SU distribution is that it can be used to estimate the probability density function of a given dataset without having prior knowledge of the underlying distribution. This estimation capability makes it useful in outlier detection, where it helps in identifying the tails of the distribution that do not conform to the general trend in the data. The Johnson SU distribution is also useful when it is not clear which distribution to choose for a given dataset.

Real world applications include:

- Aerospace engineering [6],
- Atmospheric chemistry [7],
- Bioinformatics [8],
- Biomechanics [9],
- Biomedical engineering [10],
- Climate modeling [11],
- Econometrics [12],
- Engineering [13],
- Forest science [14],
- Management science [15],
- Materials science [16],
- Occupational hygiene [17],
- Psychometrics [18],
- Remote sensing [19].

In conclusion, understanding the Johnson SU distribution is crucial in statistical inference, especially when analyzing observational data. Its flexibility and capability to handle different shapes and types of data make it a perfect probability distribution for estimating probability density functions. It is a reliable distribution with numerous applications in many fields. When you need to estimate the distribution of your data without assuming a specific form, the Johnson SU distribution can be a highly effective method for your analysis.

## References

[1] Fuzzyrandom, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

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

[3] Hossain, M. et al. (2019). Forecasting the General Index of Dhaka Stock Exchange. International Research Journal of Finance and Economics. January-February.

[4] Tsai, Cindy Sin-Yi (2011). “The Real World is Not Normal” (PDF). Morningstar Alternative Investments Observer.

[5] Choi, P. (2001). Estimation of value at risk using Johnson’s S_{U}-normal distribution. Retrieved Januay 8, 2022 from: http://zoonek.free.fr/blosxom/Finance/2007-10-28_Articles.pdf

[6] Tielrooij, M., C. Borst, M. M. van Paassen, and M. Mulder. 2015. Predicting arrival time uncertainty from actual flight information. Eleventh USA/Europe Air Traffic Management Research and Development Seminar (ATM2015); 10 pp.

[7] Mage, D. T. 1980. An explicit solution for SB parameters using four percentile points. Technometrics 22(2): 247-251.

[8] George, F., and K. M. Ramachandran. 2008. A mixture model approach for gene selection using Johnson’s system and Bayes formula. Neural, Parallel, & Scientific Computations 16: 45–58.

[9] Stanfield, P. M., J. R. Wilson, G. A. Mirka, N. F. Glasscock, J. P. Psihogios, and J. R. Davis. 1996. Multivariate input modeling with Johnson distributions. Proceedings of the 1996 Winter Simulation Conference; 8 pp.

[10] Breton, M. and B. Kovatchev. 2008. Analysis, modeling, and simulation of the accuracy of continuous glucose sensors. J. Diabetes Sci. Technol. 2(5): 853-862.

[11] Liu, F. 2012. Development and calibration of central pressure filling rate models for hurricane simulation. Unpublished M.S. Thesis, Clemson University; 130 pp.

[12] Lu, Y., O. A. Ramirez, R. M. Rejesus, T. O. Knight, and B. J. Sherrick. 2008. Empirically evaluating the flexibility of the Johnson family of distributions: a crop insurance application. Agricultural & Resource Economics Review 37(1): 79-91.

[13] Farnum, N. R. 1996. Using Johnson curves to describe non-normal process data. Quality Engineering 9(2): 329-336.

[14] Hafley, W. L. and H. T. Schreuder. 1977. Statistical distributions for fitting diameter and height data in even-aged stands. Can. J. For. Res. 7: 481-487.

[15] Alexopoulos, C., D. Goldsman, J. Fontanesi, D. Kopald, and J. R. Wilson. 2008. Modeling patient arrivals in community clinics. Omega 36: 33-43.

[16] Matthews, J. L., E. K. Lada, L. M. Weiland, R. C. Smith, and D. J. Leo. 2006. Monte Carlo simulation of a solvated ionic polymer with cluster morphology. Smart Mater. Struct. 15: 187–199.

[17] Flynn, M. R. 2007. Analysis of exposure–biomarker relationships with the Johnson SBB distribution. Ann. Occup. Hyg. 51(6): 533–541.

[18] van den Oord, E. J. C. G. 2005. Estimating Johnson curve population distributions in MULTILOG. Applied Psychological Measurement 29(1): 45–64.

[19] Ben-David, A. and C. E. Davidson. 2012. Probability theory for 3-layer remote sensing radiative transfer model: univariate case. Opt. Express 20(9): 10004-10033..