Bounded and unbounded Johnson distributions

This package demonstrates how to calculate statistical functionals of bounded ((Subscript[S, B])) and unbounded ((Subscript[S, U])) Johnson distributions.

Author:  Sergej V. Aksenov

Department of Microbiology and Immunology
University of Michigan
Ann Arbor, MI  48109, USA
aksenov@mac.com

Last updated 1 May 2002

Installation and initialization

The package file JohnsonDistributions.m should be located inside the JohnsonDistributions directory.  
The JohnsonDistributions directory should be put in one of the canonical places for AddOns.   
Two recommended places are first inside the Mathematica distribution in:

ToFileName[ {$TopDirectory,  "AddOns",  "Applications"}]

a second in the preferences directory:

\(ToFileName[\ {$PreferencesDirectory, \ "AddOns", \ "Applications"}]\)

When one of these locations is used there is no need to set $Path and the loading instructions are identical for all versions of Mathematica.

The  JohnsonDistributions` package can  be loaded with

<<JohnsonDistributions`JohnsonDistributions`

Introduction

Johnson distributions are defined by the following formulas:

Assume that z is distributed as Normal(0,1) then

x = sinh (((z - g)/d))

is distributed as unbounded Johnson variate with shape parameters g and d (of Johnson system (Subscript[S, U])),

and

x = 
      (1/(1 + (e^(-((z - g)/d)))))

is distributed as bounded Johnson variate with shape parameters g and d (of Johnson system (Subscript[S, B])).

Reference:

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

This package provides extension of standard Mathematica functions PDF, CDF, Quantile, ParameterQ, Domain, DomainQ, Random, RandomArray, Mean, Variance, StandardDeviation, Skewness, and Kurtosis to Unbounded and Bounded Johnson distributions.

Examples

Example 1

In Example 1 we demonstrate calculations with Unbounded Johnson distribution.

?UnboundedJohnsonDistribution

Domain[UnboundedJohnsonDistribution[1, 2]]

PDF[UnboundedJohnsonDistribution[1, 2], -2]

CDF[UnboundedJohnsonDistribution[1, 2], -2]

Quantile[UnboundedJohnsonDistribution[1, 2], 0.029]

SeedRandom[231445]

Random[UnboundedJohnsonDistribution[1, 2]]

SeedRandom[331908]

RandomArray[UnboundedJohnsonDistribution[1, 2], 20]

Mean[UnboundedJohnsonDistribution[1, 2]]

Variance[UnboundedJohnsonDistribution[1, 2]]

StandardDeviation[UnboundedJohnsonDistribution[1, 2]]

Skewness[UnboundedJohnsonDistribution[1, 2]]

Kurtosis[UnboundedJohnsonDistribution[1, 2]]

Example 2

In Example 2 we demonstrate calculations with Bounded Johnson distribution.

?BoundedJohnsonDistribution

Domain[BoundedJohnsonDistribution[1, 2]]

DomainQ[BoundedJohnsonDistribution[1, 2], 0.5]

DomainQ[BoundedJohnsonDistribution[1, 2], -0.5]

PDF[BoundedJohnsonDistribution[1, 2], 0.1]

CDF[BoundedJohnsonDistribution[1, 2], 0.1]

Quantile[BoundedJohnsonDistribution[1, 2], 0.1]

SeedRandom[309183]

Random[BoundedJohnsonDistribution[1, 2]]

SeedRandom[509341]

RandomArray[BoundedJohnsonDistribution[1, 2], 20]

Mean[BoundedJohnsonDistribution[1, 2]]

Variance[BoundedJohnsonDistribution[1, 2]]

StandardDeviation[BoundedJohnsonDistribution[1, 2]]

Skewness[BoundedJohnsonDistribution[1, 2]]

Kurtosis[BoundedJohnsonDistribution[1, 2]]