SSJ
V. 2.6.

umontreal.iro.lecuyer.probdist
Class JohnsonSLDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by umontreal.iro.lecuyer.probdist.JohnsonSLDist
All Implemented Interfaces:
Distribution

public class JohnsonSLDist
extends ContinuousDistribution

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

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

and distribution function

F(x) = Φ(z),        for ξ < x < ∞,

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

F-1(u) = ξ + λev(u),        for 0 <= u <= 1,

where

v(u) = [Φ-1(u) - γ]/δ.

Without loss of generality, one may choose γ = 0 or λ = 1.


Field Summary
 
Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
decPrec
 
Constructor Summary
JohnsonSLDist(double gamma, double delta)
          Same as JohnsonSLDist (gamma, delta, 0, 1).
JohnsonSLDist(double gamma, double delta, double xi, double lambda)
          Constructs a JohnsonSLDist 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 function 1 - F(x).
 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 F(x).
 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 f (x).
 double getDelta()
          Returns the value of δ.
 double getGamma()
          Returns the value of γ.
static JohnsonSLDist getInstanceFromMLE(double[] x, int n)
          Creates a new instance of a Johnson SL distribution with parameters 0, δ, ξ and λ over the interval [ξ,∞] estimated 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 SL distribution with parameters γ, δ, ξ and λ.
static double[] getMLE(double[] x, int n)
          Estimates the parameters (γ, δ, ξ, λ) of the Johnson SL 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 SL 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 SL 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 distribution function F-1(u).
 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

JohnsonSLDist

public JohnsonSLDist(double gamma,
                     double delta)
Same as JohnsonSLDist (gamma, delta, 0, 1).


JohnsonSLDist

public JohnsonSLDist(double gamma,
                     double delta,
                     double xi,
                     double lambda)
Constructs a JohnsonSLDist 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 f (x).


cdf

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


barF

public static double barF(double gamma,
                          double delta,
                          double xi,
                          double lambda,
                          double x)
Returns the complementary distribution function 1 - F(x).


inverseF

public static double inverseF(double gamma,
                              double delta,
                              double xi,
                              double lambda,
                              double u)
Returns the inverse distribution function F-1(u).


getMLE

public static double[] getMLE(double[] x,
                              int n)
Estimates the parameters (γ, δ, ξ, λ) of the Johnson SL distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a 4-element array in the order [0, δ, ξ, λ] (with γ always set to 0).

Parameters:
x - the list of observations to use to evaluate parameters
n - the number of observations to use to evaluate parameters
Returns:
returns the parameters [0, δ, ξ, λ]

getInstanceFromMLE

public static JohnsonSLDist getInstanceFromMLE(double[] x,
                                               int n)
Creates a new instance of a Johnson SL distribution with parameters 0, δ, ξ and λ over the interval [ξ,∞] estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.

Parameters:
x - the list of observations to use to evaluate parameters
n - the number of observations to use to evaluate parameters

getMean

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

Returns:
the mean of the Johnson SL distribution E[X] = ξ + λe1/2δ2-γ/δ

getVariance

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

Returns:
the variance of the Johnson SL distribution Var[X] = λ2(e1/δ2 -1)e1/δ2-2γ/δ

getStandardDeviation

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

Returns:
the standard deviation of the Johnson SL 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.