Extends the class ContinuousDistribution for the Johnson. More...

Inheritance diagram for umontreal.ssj.probdist.JohnsonSBDist:

Public Member Functions
	JohnsonSBDist (double gamma, double delta, double xi, double lambda)
	Constructs a JohnsonSBDist object with shape parameters.
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
void	setParams (double gamma, double delta, double xi, double lambda)
	Sets the value of the parameters \(\gamma\), \(\delta\),.
Public Member Functions inherited from umontreal.ssj.probdist.JohnsonSystem
double	getGamma ()
	Returns the value of \(\gamma\).
double	getDelta ()
	Returns the value of \(\delta\).
double	getXi ()
	Returns the value of \(\xi\).
double	getLambda ()
	Returns the value of \(\lambda\).
double[]	getParams ()
	Return an array containing the parameters of the current distribution.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (double gamma, double delta, double xi, double lambda, double x)
	Returns the density function ( `JohnsonSB-density` ).
static double	cdf (double gamma, double delta, double xi, double lambda, double x)
	Returns the distribution function ( `JohnsonSB-dist` ).
static double	barF (double gamma, double delta, double xi, double lambda, double x)
	Returns the complementary distribution.
static double	inverseF (double gamma, double delta, double xi, double lambda, double u)
	Returns the inverse of the distribution ( `JohnsonSB-inverse` ).
static double[]	getMLE (double[] x, int n, double xi, double lambda)
	Estimates the parameters \((\gamma,\delta)\) of the Johnson.
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 \(x[i]\), \(i = 0, 1, …, n-1\).
static double	getMean (double gamma, double delta, double xi, double lambda)
	Returns the mean [97] of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\),.
static double	getVariance (double gamma, double delta, double xi, double lambda)
	Returns the variance [62] of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\),.
static double	getStandardDeviation (double gamma, double delta, double xi, double lambda)
	Returns the standard deviation of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\),.

Additional Inherited Members
Protected Member Functions inherited from umontreal.ssj.probdist.JohnsonSystem
	JohnsonSystem (double gamma, double delta, double xi, double lambda)
	Constructs a JohnsonSystem object with shape parameters.
void	setParams0 (double gamma, double delta, double xi, double lambda)
	Sets the value of the parameters \(\gamma\), \(\delta\),.

Detailed Description

Extends the class ContinuousDistribution for the Johnson.

\(S_B\) distribution [97], [114], [62] with shape parameters \(\gamma\) and \(\delta> 0\), location parameter \(\xi\), and scale parameter \(\lambda>0\). Denoting \(t=(x-\xi)/\lambda\) and \(z = \gamma+ \delta\ln(t/(1-t))\), the density is

\[ f(x) = \frac{\delta e^{-z^2/2}}{\lambda t(1 - t)\sqrt{2\pi}}, \qquad\mbox{ for } \xi< x < \xi+\lambda, \tag{JohnsonSB-density} \]

and 0 elsewhere. The distribution function is

\[ F(x) = \Phi(z), \mbox{ for } \xi< x < \xi+\lambda, \tag{JohnsonSB-dist} \]

where \(\Phi\) is the standard normal distribution function. The inverse distribution function is

\[ F^{-1}(u) = \xi+ \lambda\left(1/\left(1+e^{-v(u)}\right)\right) \qquad\mbox{for } 0 \le u \le1, \tag{JohnsonSB-inverse} \]

where

\[ v(u) = [\Phi^{-1}(u) - \gamma]/\delta. \]

This class relies on the methods NormalDist.cdf01 and NormalDist.inverseF01 of NormalDist to approximate \(\Phi\) and \(\Phi^{-1}\).

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Definition at line 57 of file JohnsonSBDist.java.

Constructor & Destructor Documentation

◆ JohnsonSBDist()

umontreal.ssj.probdist.JohnsonSBDist.JohnsonSBDist	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda )

Constructs a JohnsonSBDist object with shape parameters.

\(\gamma\) and \(\delta\), location parameter \(\xi\) and scale parameter \(\lambda\).

Definition at line 129 of file JohnsonSBDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.JohnsonSBDist.barF	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda,
		double	x )

static

Returns the complementary distribution.

Definition at line 204 of file JohnsonSBDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.JohnsonSBDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 147 of file JohnsonSBDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.JohnsonSBDist.cdf	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda,
		double	x )

static

Returns the distribution function ( JohnsonSB-dist ).

Definition at line 187 of file JohnsonSBDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.JohnsonSBDist.cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements umontreal.ssj.probdist.Distribution.

Definition at line 143 of file JohnsonSBDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.JohnsonSBDist.density	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda,
		double	x )

static

Returns the density function ( JohnsonSB-density ).

Definition at line 171 of file JohnsonSBDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.JohnsonSBDist.density ( double x )

Returns \(f(x)\), the density evaluated at \(x\).

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 139 of file JohnsonSBDist.java.

◆ getInstanceFromMLE()

JohnsonSBDist umontreal.ssj.probdist.JohnsonSBDist.getInstanceFromMLE	(	double[]	x,
		int	n,
		double	xi,
		double	lambda )

static

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 \(\xi= \mathtt{xi}\) and

\(\lambda= \mathtt{lambda}\), the parameters \(\gamma\) and \(\delta\) 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 \(\xi\)
lambda	parameter \(\lambda\)

Definition at line 315 of file JohnsonSBDist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.JohnsonSBDist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 155 of file JohnsonSBDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.JohnsonSBDist.getMean	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda )

static

Returns the mean [97] of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\),.

\(\xi\) and \(\lambda\).

Definition at line 326 of file JohnsonSBDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.JohnsonSBDist.getMLE	(	double[]	x,
		int	n,
		double	xi,
		double	lambda )

static

Estimates the parameters \((\gamma,\delta)\) of the Johnson.

\(S_B\) distribution, using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). Parameters \(\xi= \mathtt{xi}\) and \(\lambda= \mathtt{lambda}\) are known. The estimated parameters are returned in a two-element array in the order: [ \(\gamma\), \(\delta\)]. The maximum likelihood estimators are the values \((\hat{\gamma}, \hat{\delta})\) that satisfy the equations [62] \(\hat{\gamma}= -\bar{f} / s_f\) and \(\hat{\delta}= 1/s_f\), where \(f = \ln(t/(1-t))\), \(\bar{f}\) is the sample mean of the \(f_i\), and

\[ s_f = \sqrt{\frac{1}{n} \sum_{i=0}^{n-1} (f_i - \bar{f})^2}, \]

with \(f_i = \ln(t_i/(1-t_i))\).

Parameters

x	the list of observations to use to evaluate parameters
n	the number of observations to use to evaluate parameters
xi	parameter \(\xi\)
lambda	parameter \(\lambda\)

Returns: returns the parameters [ \(\hat{\gamma}\), \(\hat{\delta}\)]

Definition at line 269 of file JohnsonSBDist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.JohnsonSBDist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 163 of file JohnsonSBDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.JohnsonSBDist.getStandardDeviation	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda )

static

Returns the standard deviation of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\),.

\(\lambda\).

Returns: the standard deviation of the Johnson \(S_B\) distribution

Definition at line 416 of file JohnsonSBDist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.JohnsonSBDist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 159 of file JohnsonSBDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.JohnsonSBDist.getVariance	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda )

static

Returns the variance [62] of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\),.

\(\xi\) and \(\lambda\).

Returns: the variance of the Johnson \(S_B\) distribution.

Definition at line 343 of file JohnsonSBDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.JohnsonSBDist.inverseF	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda,
		double	u )

static

Returns the inverse of the distribution ( JohnsonSB-inverse ).

Definition at line 222 of file JohnsonSBDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.JohnsonSBDist.inverseF ( double u )

Returns the inverse distribution function \(x = F^{-1}(u)\).

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 151 of file JohnsonSBDist.java.

◆ setParams()

void umontreal.ssj.probdist.JohnsonSBDist.setParams	(	double	gamma,
		double	delta,
		double	xi,
		double	lambda )

Sets the value of the parameters \(\gamma\), \(\delta\),.

\(\xi\) and \(\lambda\) for this object.

Definition at line 425 of file JohnsonSBDist.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/probdist/JohnsonSBDist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ JohnsonSBDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()