SSJ  3.3.1
Stochastic Simulation in Java
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JohnsonSLDist Class Reference

Extends the class ContinuousDistribution for the Johnson \(S_L\) distribution (see [101], [99] ). More...

Inheritance diagram for JohnsonSLDist:
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Collaboration diagram for JohnsonSLDist:
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Public Member Functions

 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 \(\gamma\) 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 \(f(x)\).
 
static double cdf (double gamma, double delta, double xi, double lambda, double x)
 Returns the distribution function \(F(x)\).
 
static double barF (double gamma, double delta, double xi, double lambda, double x)
 Returns the complementary distribution function \(1-F(x)\).
 
static double inverseF (double gamma, double delta, double xi, double lambda, double u)
 Returns the inverse distribution function \(F^{-1}(u)\).
 
static double [] getMLE (double[] x, int n)
 Estimates the parameters \((\gamma\), \(\delta\), \(\xi\), \(\lambda)\) of the Johnson \(S_L\) distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...
 
static JohnsonSLDist getInstanceFromMLE (double[] x, int n)
 Creates a new instance of a Johnson \(S_L\) distribution with parameters 0, \(\delta\), \(\xi\) and \(\lambda\) over the interval \([\xi,\infty]\) estimated 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

\[ E[X] = \xi+ \lambda e^{1/2\delta^2 - \gamma/\delta} \]

of the Johnson \(S_L\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\). More...

 
static double getVariance (double gamma, double delta, double xi, double lambda)
 Returns the variance

\[ \mbox{Var}[X] = \lambda^2 \left(e^{1/\delta^2} - 1\right) e^{1/\delta^2 - 2\gamma/\delta} \]

of the Johnson \(S_L\) 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 \(S_L\) 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 \(S_L\) distribution (see [101], [99] ).

It has shape parameters \(\gamma\) and \(\delta> 0\), location parameter \(\xi\), and scale parameter \(\lambda> 0\). Denoting \(t=(x-\xi)/\lambda\) and \(z = \gamma+ \delta\ln(t)\), the distribution has density

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

and distribution function

\[ F(x) = \Phi(z), \qquad\mbox{for } \xi< x < \infty, \]

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

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

where

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

Without loss of generality, one may choose \(\gamma= 0\) or \(\lambda=1\).

Member Function Documentation

◆ barF()

double barF ( double  x)

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

Parameters
xvalue 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
xvalue at which the distribution function is evaluated
Returns
distribution function evaluated at x

Implements Distribution.

◆ getInstanceFromMLE()

static JohnsonSLDist getInstanceFromMLE ( double []  x,
int  n 
)
static

Creates a new instance of a Johnson \(S_L\) distribution with parameters 0, \(\delta\), \(\xi\) and \(\lambda\) over the interval \([\xi,\infty]\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).

Parameters
xthe list of observations to use to evaluate parameters
nthe number of observations to use to evaluate parameters

◆ getMean()

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

Returns the mean

\[ E[X] = \xi+ \lambda e^{1/2\delta^2 - \gamma/\delta} \]

of the Johnson \(S_L\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\).

Returns
the mean of the Johnson \(S_L\) distribution \(E[X] = \xi+ \lambda e^{1/2\delta^2 - \gamma/\delta}\)

◆ getMLE()

static double [] getMLE ( double []  x,
int  n 
)
static

Estimates the parameters \((\gamma\), \(\delta\), \(\xi\), \(\lambda)\) 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, \(\delta\), \(\xi\), \(\lambda\).

Parameters
xthe list of observations to use to evaluate parameters
nthe number of observations to use to evaluate parameters
Returns
returns the parameters [0, \(\delta\), \(\xi\), \(\lambda\)]

◆ getStandardDeviation()

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

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

Returns
the standard deviation of the Johnson \(S_L\) distribution

◆ getVariance()

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

Returns the variance

\[ \mbox{Var}[X] = \lambda^2 \left(e^{1/\delta^2} - 1\right) e^{1/\delta^2 - 2\gamma/\delta} \]

of the Johnson \(S_L\) distribution with parameters \(\gamma\), \(\delta\), \(\xi\) and \(\lambda\).

Returns
the variance of the Johnson \(S_L\) distribution \(\mbox{Var}[X] = \lambda^2 \left(e^{1/\delta^2} - 1\right) e^{1/\delta^2 - 2\gamma/\delta}\)

◆ inverseF()

double inverseF ( double  u)

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

Parameters
uvalue 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: