Johnson Distributions

The Johnson family of distributions, first investigated by Norman L. Johnson in 1949, is based on transformations of the normal distribution. It includes four distinct types: normal, lognormal, bounded, and unbounded.

The Johnson family of distributions is defined by transformations of the normal distribution. Each variant has its own Probability Density Function (PDF) and Cumulative Distribution Function (CDF).

Normal (S_N)

The identity transformation, equivalent to the normal distribution:

$$f(x)=\frac{1}{\lambda \sqrt{2\pi }}\exp (-\frac{1}{2}{\left(\frac{x-\xi }{\lambda }\right)}^2)$$$$F(x)=\mathrm{\Phi }\left(\frac{x-\xi }{\lambda }\right)$$

Unbounded (S_U)

Based on a hyperbolic sine transformation:

$$f(x)=\frac{\delta }{\lambda \sqrt{2\pi }}\frac{1}{\sqrt{1+((x-\xi )/\lambda {)}^2}}\exp (-\frac{1}{2}{(\gamma +\delta {\sinh }^{-1}\left(\frac{x-\xi }{\lambda }\right))}^2)$$$$F(x)=\mathrm{\Phi }(\gamma +\delta {\sinh }^{-1}\left(\frac{x-\xi }{\lambda }\right))$$

Bounded (S_B)

Based on a logistic transformation, bounded between $\xi$ and $\xi +\lambda$:

$$f(x)=\frac{\delta }{\lambda \sqrt{2\pi }}\frac{1}{(x-\xi )(1-(x-\xi )/\lambda )}\exp (-\frac{1}{2}{(\gamma +\delta \ln \left(\frac{x-\xi }{\lambda -(x-\xi )}\right))}^2)$$$$F(x)=\mathrm{\Phi }(\gamma +\delta \ln \left(\frac{x-\xi }{\lambda -(x-\xi )}\right))$$

LogNormal (S_L)

Based on an exponential transformation, bounded on the left by $\xi$:

$$f(x)=\frac{\delta }{\lambda \sqrt{2\pi }}\frac{1}{x-\xi }\exp (-\frac{1}{2}{(\gamma +\delta \ln \left(\frac{x-\xi }{\lambda }\right))}^2)$$$$F(x)=\mathrm{\Phi }(\gamma +\delta \ln \left(\frac{x-\xi }{\lambda }\right))$$

All Johnson distributions have a location parameter $\xi$ and a scale parameter $\lambda$. The bounded, unbounded, and lognormal variants have two additional shape parameters $\gamma$ and $\delta$. Here, $\mathrm{\Phi }$ represents the standard normal CDF.

Johnson distributions are used in various fields such as finance, hydrology, and environmental science to model data that do not fit well with normal distributions. They are particularly useful for modeling skewed and bounded data.

The S_B bounded distribution is commonly used in quality control to model manufacturing processes with natural upper and lower limits. The S_U unbounded distribution finds applications in financial modeling where data can be highly skewed but theoretically unbounded.

The S_L lognormal variant is frequently applied in environmental monitoring to model pollutant concentrations, which cannot be negative but can have large positive values. In reliability engineering, Johnson distributions are used to model failure times and material strength distributions.

The statistical properties depend on the specific Johnson distribution type. For the S_N type, see the normal distribution.

For the S_U (Unbounded) type:

For the S_L (LogNormal) type:

For the S_B (Bounded) type:

Notable properties include:

• All Johnson distributions are transformations of the standard normal distribution.

• The S_B type is bounded on both sides by $\xi$ and $\xi +\lambda$.

• The S_L type is bounded on the left by $\xi$.

• The S_N type is exactly the normal distribution.

• The S_L type includes the lognormal distribution as a special case.

• The S_B type includes the logistic distribution as a limiting case.

The Johnson family of distributions is implemented by the JohnsonDistribution class. It has two constructors. The first constructor takes five arguments. These are the location and scale parameters, the two shape parameters, and a JohnsonDistributionType value that specifies which variant is being defined.

The following constructs an S_U distribution with location parameter 6.8, scale parameter 4.1, and shape parameters 1.3 and 3.2:

var johnsonu = new JohnsonDistribution(6.8, 4.1, 1.3, 3.2, 
    JohnsonDistributionType.Unbounded);

Dim johnsonu = new JohnsonDistribution(6.8, 4.1, 1.3, 3.2, 
    JohnsonDistributionType.Unbounded)

No code example is currently available or this language may not be supported.

let johnsonu = new JohnsonDistribution(6.8, 4.1, 1.3, 3.2, 
    JohnsonDistributionType.Unbounded)

The second constructor takes a vector argument and estimates the distribution from the provided sample:

var sample = Vector.Create(1.2, 3.2, 4.1, 6.1, 2.3, 1.0, 2.1);
var johnson = new JohnsonDistribution(sample);

Dim sample = Vector.Create(1.2, 3.2, 4.1, 6.1, 2.3, 1.0, 2.1)
Dim johnson = new JohnsonDistribution(sample)

No code example is currently available or this language may not be supported.

let sample = Vector.Create(1.2, 3.2, 4.1, 6.1, 2.3, 1.0, 2.1)
let johnson = new JohnsonDistribution(sample)

Note that parameter estimation says nothing about how well the estimated distribution fits the variable's distribution. Use one of the goodness-of-fit tests to verify the appropriateness of the choice of distribution.

The JohnsonDistribution class has five specific properties. LocationParameter and ScaleParameter return the location and scale parameters of the distribution. Gamma and Delta return the shape parameters, and Type returns the type of Johnson distribution.

• Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36(1/2), 149-176.

• Hill, I. D., Hill, R., & Holder, R. L. (1976). Fitting Johnson curves by moments. Applied Statistics, 180-189.

Other Resources