This class has been replaced by GumbelDist . More...

Inheritance diagram for umontreal.ssj.probdist.ExtremeValueDist:

Public Member Functions
	ExtremeValueDist ()
	THIS CLASS HAS BEEN REPLACED BY GumbelDist .
	ExtremeValueDist (double alpha, double lambda)
	THIS CLASS HAS BEEN REPLACED BY GumbelDist .
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.
double	getAlpha ()
	Returns the parameter \(\alpha\) of this object.
double	getLambda ()
	Returns the parameter \(\lambda\) of this object.
void	setParams (double alpha, double lambda)
	Sets the parameters \(\alpha\) and \(\lambda\) of this object.
double[]	getParams ()
	Return a table 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 alpha, double lambda, double x)
	Computes the density function.
static double	cdf (double alpha, double lambda, double x)
	THIS CLASS HAS BEEN REPLACED BY GumbelDist .
static double	barF (double alpha, double lambda, double x)
	Computes the complementary distribution function.
static double	inverseF (double alpha, double lambda, double u)
	Computes the inverse distribution function.
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha,\lambda)\) of the extreme value distribution using the maximum likelihood method, from the.
static double[]	getMaximumLikelihoodEstimate (double[] x, int n)
	Same as getMLE.
static ExtremeValueDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of an extreme value distribution with parameters \(\alpha\) and \(\lambda\) estimated using the maximum likelihood method based on the \(n\) observations.
static double	getMean (double alpha, double lambda)
	Computes and returns the mean, \(E[X] = \alpha+ \gamma/\lambda\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\), where \(\gamma= 0.5772156649\) is the Euler-Mascheroni constant.
static double	getVariance (double alpha, double lambda)
	Computes and returns the variance, \(\mbox{Var}[X] = \pi^2/(6\lambda^2)\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\).
static double	getStandardDeviation (double alpha, double lambda)
	Computes and returns the standard deviation of the extreme value distribution with parameters \(\alpha\) and \(\lambda\).

Detailed Description

This class has been replaced by GumbelDist .

Extends the class ContinuousDistribution for the extreme value (or Gumbel) distribution [96] (page 2), with location parameter \(\alpha\) and scale parameter \(\lambda> 0\). It has density

\[ f (x) = \lambda e^{-\lambda(x-\alpha)} e^{-e^{-\lambda(x-\alpha)}}, \qquad\qquad\mbox{for } -\infty< x < \infty, \tag{fextremevalue} \]

distribution function

\[ F(x) = e^{-e^{-\lambda(x - \alpha)}} \qquad\qquad\mbox{for } -\infty< x < \infty, \tag{Fextreme} \]

and inverse distribution function

\[ F^{-1}(u) = -\ln(-\ln(u))/\lambda+ \alpha, \qquad\mbox{for } 0 \le u \le1. \]

Definition at line 51 of file ExtremeValueDist.java.

Constructor & Destructor Documentation

◆ ExtremeValueDist() [1/2]

umontreal.ssj.probdist.ExtremeValueDist.ExtremeValueDist ( )

THIS CLASS HAS BEEN REPLACED BY GumbelDist .

Constructs a ExtremeValueDist object with parameters \(\alpha\) = 0 and \(\lambda\) = 1.

Definition at line 90 of file ExtremeValueDist.java.

◆ ExtremeValueDist() [2/2]

umontreal.ssj.probdist.ExtremeValueDist.ExtremeValueDist	(	double	alpha,
		double	lambda )

THIS CLASS HAS BEEN REPLACED BY GumbelDist .

Constructs a ExtremeValueDist object with parameters \(\alpha\) = alpha and \(\lambda\) = lambda.

Definition at line 99 of file ExtremeValueDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.ExtremeValueDist.barF	(	double	alpha,
		double	lambda,
		double	x )

static

Computes the complementary distribution function.

Definition at line 162 of file ExtremeValueDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.ExtremeValueDist.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 111 of file ExtremeValueDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.ExtremeValueDist.cdf	(	double	alpha,
		double	lambda,
		double	x )

static

THIS CLASS HAS BEEN REPLACED BY GumbelDist .

Computes the distribution function.

Definition at line 148 of file ExtremeValueDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.ExtremeValueDist.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 107 of file ExtremeValueDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.ExtremeValueDist.density	(	double	alpha,
		double	lambda,
		double	x )

static

Computes the density function.

Definition at line 134 of file ExtremeValueDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.ExtremeValueDist.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 103 of file ExtremeValueDist.java.

◆ getAlpha()

double umontreal.ssj.probdist.ExtremeValueDist.getAlpha ( )

Returns the parameter \(\alpha\) of this object.

Definition at line 310 of file ExtremeValueDist.java.

◆ getInstanceFromMLE()

ExtremeValueDist umontreal.ssj.probdist.ExtremeValueDist.getInstanceFromMLE	(	double[]	x,
		int	n )

static

Creates a new instance of an extreme value distribution with parameters \(\alpha\) and \(\lambda\) 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

Definition at line 259 of file ExtremeValueDist.java.

◆ getLambda()

double umontreal.ssj.probdist.ExtremeValueDist.getLambda ( )

Returns the parameter \(\lambda\) of this object.

Definition at line 317 of file ExtremeValueDist.java.

◆ getMaximumLikelihoodEstimate()

double[] umontreal.ssj.probdist.ExtremeValueDist.getMaximumLikelihoodEstimate	(	double[]	x,
		int	n )

static

Same as getMLE.

Definition at line 246 of file ExtremeValueDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 119 of file ExtremeValueDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.ExtremeValueDist.getMean	(	double	alpha,
		double	lambda )

static

Computes and returns the mean, \(E[X] = \alpha+ \gamma/\lambda\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\), where \(\gamma= 0.5772156649\) is the Euler-Mascheroni constant.

Returns: the mean of the Extreme Value distribution \(E[X] = \alpha+ \gamma/ \lambda\)

Definition at line 272 of file ExtremeValueDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.ExtremeValueDist.getMLE	(	double[]	x,
		int	n )

static

Estimates the parameters \((\alpha,\lambda)\) of the extreme value distribution using the maximum likelihood method, from the.

\(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). The estimates are returned in a two-element array, in regular order: [ \(\alpha\), \(\lambda\)]. The maximum likelihood estimators are the values \((\hat{\alpha}, \hat{\lambda})\) that satisfy the equations:

\begin{align*} \hat{\lambda} & = \bar{x}_n - \frac{\sum_{i=1}^n x_i e^{- \hat{\lambda} x_i}}{\sum_{i=1}^n e^{-\hat{\lambda} x_i}} \\ \hat{\alpha} & = - \frac{1}{\hat{\lambda}} \ln\left( \frac{1}{n} \sum_{i=1}^n e^{-\hat{\lambda} x_i} \right), \end{align*}

where \(\bar{x}_n\) is the average of \(x[0],…,x[n-1]\) [56] (page 89).

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters

Returns: returns the parameters [ \(\hat{\alpha}\), \(\hat{\lambda}\)]

Definition at line 209 of file ExtremeValueDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.ExtremeValueDist.getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha\), \(\lambda\)].

Implements umontreal.ssj.probdist.Distribution.

Definition at line 335 of file ExtremeValueDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 127 of file ExtremeValueDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.ExtremeValueDist.getStandardDeviation	(	double	alpha,
		double	lambda )

static

Computes and returns the standard deviation of the extreme value distribution with parameters \(\alpha\) and \(\lambda\).

Returns: the standard deviation of the extreme value distribution

Definition at line 300 of file ExtremeValueDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 123 of file ExtremeValueDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.ExtremeValueDist.getVariance	(	double	alpha,
		double	lambda )

static

Computes and returns the variance, \(\mbox{Var}[X] = \pi^2/(6\lambda^2)\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\).

Returns: the variance of the extreme value distribution \(\mbox{Var}[X] = 1/6 \pi^2 1/\lambda^2\)

Definition at line 287 of file ExtremeValueDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.ExtremeValueDist.inverseF	(	double	alpha,
		double	lambda,
		double	u )

static

Computes the inverse distribution function.

Definition at line 176 of file ExtremeValueDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.ExtremeValueDist.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 115 of file ExtremeValueDist.java.

◆ setParams()

void umontreal.ssj.probdist.ExtremeValueDist.setParams	(	double	alpha,
		double	lambda )

Sets the parameters \(\alpha\) and \(\lambda\) of this object.

Definition at line 324 of file ExtremeValueDist.java.

◆ toString()

String umontreal.ssj.probdist.ExtremeValueDist.toString ( )

Returns a String containing information about the current distribution.

Definition at line 343 of file ExtremeValueDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ ExtremeValueDist() [1/2]

◆ ExtremeValueDist() [2/2]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getAlpha()

◆ getInstanceFromMLE()

◆ getLambda()

◆ getMaximumLikelihoodEstimate()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()