Extends the class ContinuousDistribution for the Gumbel distribution [100] (page 2), with location parameter \(\delta\) and scale parameter \(\beta\neq0\). More...

Inheritance diagram for GumbelDist:

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Collaboration diagram for GumbelDist:

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Public Member Functions
	GumbelDist ()
	Constructor for the standard Gumbel distribution with parameters \(\beta\) = 1 and \(\delta\) = 0.

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

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.

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

static double []	getMLEmin (double[] x, int n)
	Similar to getMLE, but for the case \(\beta< 0\). More...

static GumbelDist	getInstanceFromMLE (double[] x, int n)
	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 \(\beta> 0\). More...

static GumbelDist	getInstanceFromMLEmin (double[] x, int n)
	Similar to getInstanceFromMLE, but for the case \(\beta< 0\). More...

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

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 \(\delta\). More...

static double	getStandardDeviation (double beta, double delta)
	Returns the standard deviation of the Gumbel distribution with parameters \(\beta\) and \(\delta\). More...

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

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 Gumbel distribution [100] (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

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

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 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

◆ getInstanceFromMLEmin()

static 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

◆ getMean()

static double 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 \(E[X] = \delta+ \gamma* \beta\)

◆ getMLE()

static double [] 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 [ \(\hat{\delta}\), \(\hat{\beta}\)]

◆ getMLEmin()

static double [] 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 [ \(\hat{\delta}\), \(\hat{\beta}\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

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

Implements Distribution.

◆ getStandardDeviation()

static double 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

◆ getVariance()

static double 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 \(\mbox{Var}[X] = ()\pi\beta)^2/6\)

◆ 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:

GumbelDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getInstanceFromMLEmin()

◆ getMean()

◆ getMLE()

◆ getMLEmin()

◆ getParams()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()