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.
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 
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 . That has made it useful in a variety of areas including modeling asset returns for portfolio management  and Value at Risk (VaR) modeling .
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 ,
- Atmospheric chemistry ,
- Bioinformatics ,
- Biomechanics ,
- Biomedical engineering ,
- Climate modeling ,
- Econometrics ,
- Engineering ,
- Forest science ,
- Management science ,
- Materials science ,
- Occupational hygiene ,
- Psychometrics ,
- Remote sensing .
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.
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