Johnson Distributions

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

Normal: A normal distribution, based on the identity transformation. Also referred to as the S_N form.
Unbounded: distribution unbounded on either side, based on a hyperbolic sine transformation. Also referred to as the S_U form.
Bounded: A distribution that is bounded on both sides, based on a logistic transformation. Also referred to as the S_B form.
LogNormal: A distribution similar to a lognormal distribution that is bounded on the left side, based on an exponential transformation. Also referred to as the S_L form.

All Johnson distributions have a location parameter and a scale parameter The bounded and unbounded variants have two additional shape parameters, usually denoted by γ and δ.

The Cumulative Distribution Function of the S_L variant is:

$$F(x) = \Phi\left(\gamma+\delta \sinh^{-1}\left(\frac{x-\xi}{\lambda}\right)\right)$$

where Φ is the CDF of the normal distribution. The distribution functions for the other variants are similar.

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);

Visual Basic

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

Visual Basic

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

Visual Basic

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);

Visual Basic

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

Visual Basic

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

Visual Basic

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.

For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions.