SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class JohnsonSBDist

```java.lang.Object
umontreal.iro.lecuyer.probdist.ContinuousDistribution
umontreal.iro.lecuyer.probdist.JohnsonSBDist
```
All Implemented Interfaces:
Distribution

`public class JohnsonSBDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the Johnson SB distribution with shape parameters γ and δ > 0, location parameter ξ, and scale parameter λ > 0. Denoting t = (x - ξ)/λ and z = γ + δln(t/(1 - t)), the density is

f (x) = δe-z2/2/(λt(1-t)(2π)1/2),         for ξ < x < ξ + λ,

and 0 elsewhere. The distribution function is

F(x) = Φ(z), for ξ < x < ξ + λ,

where Φ is the standard normal distribution function. The inverse distribution function is

F-1(u) = ξ + λ(1/(1 + e-v(u)))        for 0 <= u <= 1,

where

v(u) = [Φ-1(u) - γ]/δ.

This class relies on the methods `NormalDist.cdf01` and `NormalDist.inverseF01` of `NormalDist` to approximate Φ and Φ-1.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
`decPrec`

Constructor Summary
```JohnsonSBDist(double gamma, double delta, double xi, double lambda)```
Constructs a JohnsonSBDist object with shape parameters γ and δ, location parameter ξ and scale parameter λ.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double gamma, double delta, double xi, double lambda, double x)```
Returns the complementary distribution.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double gamma, double delta, double xi, double lambda, double x)```
Returns the distribution function.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double gamma, double delta, double xi, double lambda, double x)```
Returns the density function.
` double` `getDelta()`
Returns the value of δ.
` double` `getGamma()`
Returns the value of γ.
`static JohnsonSBDist` ```getInstanceFromMLE(double[] x, int n, double xi, double lambda)```
Creates a new instance of a JohnsonSBDist object using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
` double` `getLambda()`
Returns the value of λ.
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double gamma, double delta, double xi, double lambda)```
Returns the mean of the Johnson SB distribution with parameters γ, δ, ξ and λ.
`static double[]` ```getMLE(double[] x, int n, double xi, double lambda)```
Estimates the parameters (γ, δ) of the Johnson SB distribution, using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
` double[]` `getParams()`
Return an array containing the parameters of the current distribution.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` ```getStandardDeviation(double gamma, double delta, double xi, double lambda)```
Returns the standard deviation of the Johnson SB distribution with parameters γ, δ, ξ, λ.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double gamma, double delta, double xi, double lambda)```
Returns the variance of the Johnson SB distribution with parameters γ, δ, ξ and λ.
` double` `getXi()`
Returns the value of ξ.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double gamma, double delta, double xi, double lambda, double u)```
Returns the inverse of the distribution.
` void` ```setParams(double gamma, double delta, double xi, double lambda)```
Sets the value of the parameters γ, δ, ξ and λ for this object.
` String` `toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
`getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup`

Methods inherited from class java.lang.Object
`equals, getClass, hashCode, notify, notifyAll, wait, wait, wait`

Constructor Detail

### JohnsonSBDist

```public JohnsonSBDist(double gamma,
double delta,
double xi,
double lambda)```
Constructs a JohnsonSBDist object with shape parameters γ and δ, location parameter ξ and scale parameter λ.

Method Detail

### density

`public double density(double x)`
Description copied from class: `ContinuousDistribution`
Returns f (x), the density evaluated at x.

Specified by:
`density` in class `ContinuousDistribution`
Parameters:
`x` - value at which the density is evaluated
Returns:
density function evaluated at x

### cdf

`public double cdf(double x)`
Description copied from interface: `Distribution`
Returns the distribution function F(x).

Parameters:
`x` - value at which the distribution function is evaluated
Returns:
distribution function evaluated at x

### barF

`public double barF(double x)`
Description copied from class: `ContinuousDistribution`
Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:
`barF` in interface `Distribution`
Overrides:
`barF` in class `ContinuousDistribution`
Parameters:
`x` - value at which the complementary distribution function is evaluated
Returns:
complementary distribution function evaluated at x

### inverseF

`public double inverseF(double u)`
Description copied from class: `ContinuousDistribution`
Returns the inverse distribution function x = F-1(u). Restrictions: u∈[0, 1].

Specified by:
`inverseF` in interface `Distribution`
Overrides:
`inverseF` in class `ContinuousDistribution`
Parameters:
`u` - value at which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

### getMean

`public double getMean()`
Description copied from class: `ContinuousDistribution`
Returns the mean.

Specified by:
`getMean` in interface `Distribution`
Overrides:
`getMean` in class `ContinuousDistribution`
Returns:
the mean

### getVariance

`public double getVariance()`
Description copied from class: `ContinuousDistribution`
Returns the variance.

Specified by:
`getVariance` in interface `Distribution`
Overrides:
`getVariance` in class `ContinuousDistribution`
Returns:
the variance

### getStandardDeviation

`public double getStandardDeviation()`
Description copied from class: `ContinuousDistribution`
Returns the standard deviation.

Specified by:
`getStandardDeviation` in interface `Distribution`
Overrides:
`getStandardDeviation` in class `ContinuousDistribution`
Returns:
the standard deviation

### density

```public static double density(double gamma,
double delta,
double xi,
double lambda,
double x)```
Returns the density function.

### cdf

```public static double cdf(double gamma,
double delta,
double xi,
double lambda,
double x)```
Returns the distribution function.

### barF

```public static double barF(double gamma,
double delta,
double xi,
double lambda,
double x)```
Returns the complementary distribution.

### inverseF

```public static double inverseF(double gamma,
double delta,
double xi,
double lambda,
double u)```
Returns the inverse of the distribution.

### getMLE

```public static double[] getMLE(double[] x,
int n,
double xi,
double lambda)```
Estimates the parameters (γ, δ) of the Johnson SB distribution, using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. Parameters ξ = xi and λ = lambda are known. The estimated parameters are returned in a two-element array in the order: [γ, δ].

Parameters:
`x` - the list of observations to use to evaluate parameters
`n` - the number of observations to use to evaluate parameters
`xi` - parameter ξ
`lambda` - parameter λ
Returns:
returns the parameters [ hat(γ), hat(δ)]

### getInstanceFromMLE

```public static JohnsonSBDist getInstanceFromMLE(double[] x,
int n,
double xi,
double lambda)```
Creates a new instance of a JohnsonSBDist object using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1. Given the parameters ξ = xi and λ = lambda, the parameters γ and δ are estimated from the observations.

Parameters:
`x` - the list of observations to use to evaluate parameters
`n` - the number of observations to use to evaluate parameters
`xi` - parameter ξ
`lambda` - parameter λ

### getMean

```public static double getMean(double gamma,
double delta,
double xi,
double lambda)```
Returns the mean of the Johnson SB distribution with parameters γ, δ, ξ and λ.

### getVariance

```public static double getVariance(double gamma,
double delta,
double xi,
double lambda)```
Returns the variance of the Johnson SB distribution with parameters γ, δ, ξ and λ.

Returns:
the variance of the Johnson SB distribution.

### getStandardDeviation

```public static double getStandardDeviation(double gamma,
double delta,
double xi,
double lambda)```
Returns the standard deviation of the Johnson SB distribution with parameters γ, δ, ξ, λ.

Returns:
the standard deviation of the Johnson SB distribution

### setParams

```public void setParams(double gamma,
double delta,
double xi,
double lambda)```
Sets the value of the parameters γ, δ, ξ and λ for this object.

### getGamma

`public double getGamma()`
Returns the value of γ.

### getDelta

`public double getDelta()`
Returns the value of δ.

### getXi

`public double getXi()`
Returns the value of ξ.

### getLambda

`public double getLambda()`
Returns the value of λ.

### getParams

`public double[] getParams()`
Return an array containing the parameters of the current distribution. This array is put in regular order: [γ, δ, ξ, λ].

### toString

`public String toString()`
Overrides:
`toString` in class `Object`

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.