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 JohnsonSBDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Johnson S_B 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 S_B 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 S_B 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 S_B distribution with parameters γ, δ, ξ and λ.

Returns:

the variance of the Johnson S_B distribution.

getStandardDeviation

public static double getStandardDeviation(double gamma,
                                          double delta,
                                          double xi,
                                          double lambda)

Returns the standard deviation of the Johnson S_B distribution with parameters γ, δ, ξ, λ.

Returns:

the standard deviation of the Johnson S_B 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