umontreal.iro.lecuyer.probdist
Class JohnsonSLDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.JohnsonSLDist

All Implemented Interfaces:

Distribution

public class JohnsonSLDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Johnson S_L 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 + λe^v(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 S_L 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 S_L 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 S_L distribution with parameters γ, δ, ξ and λ.

Returns:

the mean of the Johnson S_L distribution E[X] = ξ + λe^1/2δ2-γ/δ

getVariance

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

Returns the variance of the Johnson S_L distribution with parameters γ, δ, ξ and λ.

Returns:

the variance of the Johnson S_L distribution Var[X] = λ²(e^1/δ2 -1)e^1/δ2-2γ/δ

getStandardDeviation

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

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

Returns:

the standard deviation of the Johnson S_L 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