Extends the class ContinuousDistribution for the Johnson \(S_B\) distribution [101], [118], [63] with shape parameters \(\gamma\) and \(\delta> 0\), location parameter \(\xi\), and scale parameter \(\lambda>0\). More...

Inheritance diagram for JohnsonSBDist:

[legend]

Collaboration diagram for JohnsonSBDist:

[legend]

Public Member Functions
	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 \(\lambda\).

double	density (double x)

double	cdf (double x)
	Returns the distribution function \(F(x)\). More...

double	barF (double x)
	Returns \(\bar{F}(x) = 1 - F(x)\). More...

double	inverseF (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More...

double	getMean ()
	Returns the mean of the distribution function.

double	getVariance ()
	Returns the variance of the distribution function.

double	getStandardDeviation ()
	Returns the standard deviation of the distribution function.

void	setParams (double gamma, double delta, double xi, double lambda)
	Sets the value of the parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\) for this object.

Public Member Functions inherited from 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. More...

String	toString ()
	Returns a `String` containing information about the current distribution.

Public Member Functions inherited from ContinuousDistribution
abstract double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\). More...

double	barF (double x)
	Returns the complementary distribution function. More...

double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...

double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...

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

double	getMean ()
	Returns the mean. More...

double	getVariance ()
	Returns the variance. More...

double	getStandardDeviation ()
	Returns the standard deviation. More...

double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXinf (double xa)
	Sets the value \(x_a=\) `xa`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXsup (double xb)
	Sets the value \(x_b=\) `xb`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

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 \(S_B\) distribution, using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...

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 \(x[i]\), \(i = 0, 1, …, n-1\). More...

static double	getMean (double gamma, double delta, double xi, double lambda)
	Returns the mean [101] of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\).

static double	getVariance (double gamma, double delta, double xi, double lambda)
	Returns the variance [63] of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\). More...

static double	getStandardDeviation (double gamma, double delta, double xi, double lambda)
	Returns the standard deviation of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\), \(\lambda\). More...

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

Protected Member Functions inherited from JohnsonSystem
	JohnsonSystem (double gamma, double delta, double xi, double lambda)
	Constructs a `JohnsonSystem` object with shape parameters \(\gamma= \mathtt{gamma}\) and \(\delta= \mathtt{delta}\), location parameter \(\xi= \mathtt{xi}\), and scale parameter \(\lambda= \mathtt{lambda}\).

void	setParams0 (double gamma, double delta, double xi, double lambda)
	Sets the value of the parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\).

Protected Attributes inherited from JohnsonSystem
double	gamma

double	delta

double	xi

double	lambda

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

Static Protected Attributes inherited from ContinuousDistribution
static final double	XBIG = 100.0

static final double	XBIGM = 1000.0

static final double []	EPSARRAY

Detailed Description

Extends the class ContinuousDistribution for the Johnson \(S_B\) distribution [101], [118], [63] 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 \(\Phi^{-1}\).

Member Function Documentation

◆ barF()

double barF ( double x )

Returns \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Implements Distribution.

◆ cdf()

double 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 Distribution.

◆ getInstanceFromMLE()

static 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 \(\lambda\)

◆ getMLE()

static double [] 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 [63] \(\hat{\gamma}= -\bar{f} / s_f\) 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 [ \(\hat{\gamma}\), \(\hat{\delta}\)]

◆ getStandardDeviation()

static double 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

◆ getVariance()

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

static

Returns the variance [63] of the Johnson \(S_B\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\).

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

◆ inverseF()

double inverseF ( double u )

Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).

Parameters

u	value in the interval \((0,1)\) for which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Implements Distribution.

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

JohnsonSBDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMLE()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()