Extends the class ContinuousDistribution for the Gumbel distribution [96] (page 2), with location parameter. More...

Inheritance diagram for umontreal.ssj.probdist.GumbelDist:

Public Member Functions
	GumbelDist ()
	Constructor for the standard Gumbel distribution with parameters.
	GumbelDist (double beta, double delta)
	Constructs a GumbelDist object with parameters \(\beta\) = beta and \(\delta\) = delta.
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	getBeta ()
	Returns the parameter \(\beta\) of this object.
double	getDelta ()
	Returns the parameter \(\delta\) of this object.
void	setParams (double beta, double delta)
	Sets the parameters \(\beta\) and \(\delta\) 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 beta, double delta, double x)
	Computes and returns the density function.
static double	cdf (double beta, double delta, double x)
	Computes and returns the distribution function.
static double	barF (double beta, double delta, double x)
	Computes and returns the complementary distribution function \(1 - F(x)\).
static double	inverseF (double beta, double delta, double u)
	Computes and returns the inverse distribution function.
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\beta,\delta)\) of the Gumbel distribution, assuming that \(\beta> 0\), and using the maximum likelihood method with the \(n\) observations \(x[i]\),.
static double[]	getMLEmin (double[] x, int n)
	Similar to getMLE, but for the case \(\beta< 0\).
static GumbelDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of an Gumbel distribution with parameters.
static GumbelDist	getInstanceFromMLEmin (double[] x, int n)
	Similar to getInstanceFromMLE, but for the case \(\beta< 0\).
static double	getMean (double beta, double delta)
	Returns the mean, \(E[X] = \delta+ \gamma\beta\), of the Gumbel distribution with parameters \(\beta\) and \(\delta\), where.
static double	getVariance (double beta, double delta)
	Returns the variance \(\mbox{Var}[X] = \pi^2 \beta^2\!/6\) of the Gumbel distribution with parameters \(\beta\) and.
static double	getStandardDeviation (double beta, double delta)
	Returns the standard deviation of the Gumbel distribution with parameters \(\beta\) and \(\delta\).

Detailed Description

Extends the class ContinuousDistribution for the Gumbel distribution [96] (page 2), with location parameter.

\(\delta\) and scale parameter \(\beta\neq0\). Using the notation \(z = (x-\delta)/\beta\), it has density

\[ f (x) = \frac{e^{-z} e^{-e^{-z}}}{|\beta|}, \qquad\mbox{for } -\infty< x < \infty \tag{densgumbel} \]

and distribution function

\[ F(x) = \left\{ \begin{array}{ll} e^{-e^{-z}}, \qquad & \mbox{for } \beta> 0 \\ 1 - e^{-e^{-z}}, \qquad & \mbox{for } \beta< 0. \end{array} \right. \]

Definition at line 46 of file GumbelDist.java.

Constructor & Destructor Documentation

◆ GumbelDist() [1/2]

umontreal.ssj.probdist.GumbelDist.GumbelDist ( )

Constructor for the standard Gumbel distribution with parameters.

\(\beta\) = 1 and \(\delta\) = 0.

Definition at line 117 of file GumbelDist.java.

◆ GumbelDist() [2/2]

umontreal.ssj.probdist.GumbelDist.GumbelDist	(	double	beta,
		double	delta )

Constructs a GumbelDist object with parameters \(\beta\) = beta and \(\delta\) = delta.

Definition at line 125 of file GumbelDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.GumbelDist.barF	(	double	beta,
		double	delta,
		double	x )

static

Computes and returns the complementary distribution function \(1 - F(x)\).

Definition at line 191 of file GumbelDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.GumbelDist.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 137 of file GumbelDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.GumbelDist.cdf	(	double	beta,
		double	delta,
		double	x )

static

Computes and returns the distribution function.

Definition at line 173 of file GumbelDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.GumbelDist.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 133 of file GumbelDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.GumbelDist.density	(	double	beta,
		double	delta,
		double	x )

static

Computes and returns the density function.

Definition at line 160 of file GumbelDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.GumbelDist.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 129 of file GumbelDist.java.

◆ getBeta()

double umontreal.ssj.probdist.GumbelDist.getBeta ( )

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

Definition at line 399 of file GumbelDist.java.

◆ getDelta()

double umontreal.ssj.probdist.GumbelDist.getDelta ( )

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

Definition at line 406 of file GumbelDist.java.

◆ getInstanceFromMLE()

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

static

Creates a new instance of an Gumbel distribution with parameters.

\(\beta\) and \(\delta\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\), assuming that \(\beta> 0\).

Parameters

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

Definition at line 338 of file GumbelDist.java.

◆ getInstanceFromMLEmin()

GumbelDist umontreal.ssj.probdist.GumbelDist.getInstanceFromMLEmin	(	double[]	x,
		int	n )

static

Similar to getInstanceFromMLE, but for the case \(\beta< 0\).

Parameters

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

Definition at line 349 of file GumbelDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 145 of file GumbelDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.GumbelDist.getMean	(	double	beta,
		double	delta )

static

Returns the mean, \(E[X] = \delta+ \gamma\beta\), of the Gumbel distribution with parameters \(\beta\) and \(\delta\), where.

\(\gamma= 0.5772156649015329\) is the Euler-Mascheroni constant.

Returns: the mean of the Extreme Value distribution \(E[X] = \delta+ \gamma* \beta\)

Definition at line 362 of file GumbelDist.java.

◆ getMLE()

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

static

Estimates the parameters \((\beta,\delta)\) of the Gumbel distribution, assuming that \(\beta> 0\), and using the maximum likelihood method with the \(n\) observations \(x[i]\),.

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

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

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

Parameters

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

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

Definition at line 243 of file GumbelDist.java.

◆ getMLEmin()

double[] umontreal.ssj.probdist.GumbelDist.getMLEmin	(	double[]	x,
		int	n )

static

Similar to getMLE, but for the case \(\beta< 0\).

Parameters

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

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

Definition at line 289 of file GumbelDist.java.

◆ getParams()

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

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\beta\), \(\delta\)].

Implements umontreal.ssj.probdist.Distribution.

Definition at line 424 of file GumbelDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 153 of file GumbelDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.GumbelDist.getStandardDeviation	(	double	beta,
		double	delta )

static

Returns the standard deviation of the Gumbel distribution with parameters \(\beta\) and \(\delta\).

Returns: the standard deviation of the Gumbel distribution

Definition at line 389 of file GumbelDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 149 of file GumbelDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.GumbelDist.getVariance	(	double	beta,
		double	delta )

static

Returns the variance \(\mbox{Var}[X] = \pi^2 \beta^2\!/6\) of the Gumbel distribution with parameters \(\beta\) and.

\(\delta\).

Returns: the variance of the Gumbel distribution \(\mbox{Var}[X] = ()\pi\beta)^2/6\)

Definition at line 376 of file GumbelDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.GumbelDist.inverseF	(	double	beta,
		double	delta,
		double	u )

static

Computes and returns the inverse distribution function.

Definition at line 209 of file GumbelDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.GumbelDist.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 141 of file GumbelDist.java.

◆ setParams()

void umontreal.ssj.probdist.GumbelDist.setParams	(	double	beta,
		double	delta )

Sets the parameters \(\beta\) and \(\delta\) of this object.

Definition at line 413 of file GumbelDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 432 of file GumbelDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ GumbelDist() [1/2]

◆ GumbelDist() [2/2]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getBeta()

◆ getDelta()

◆ getInstanceFromMLE()

◆ getInstanceFromMLEmin()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getMLEmin()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()