Extends the class ContinuousDistribution for the Fréchet distribution [100] (page 3), with location parameter \(\delta\), scale parameter \(\beta> 0\), and shape parameter \(\alpha> 0\), where we use the notation \(z = (x-\delta)/\beta\). More...

Inheritance diagram for FrechetDist:

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Collaboration diagram for FrechetDist:

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Public Member Functions
	FrechetDist (double alpha)
	Constructor for the standard Fréchet distribution with parameters \(\beta\) = 1 and \(\delta\) = 0.

	FrechetDist (double alpha, double beta, double delta)
	Constructs a `FrechetDist` object with parameters \(\alpha\) = `alpha`, \(\beta\) = `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	getAlpha ()
	Returns the parameter \(\alpha\) of this object.

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

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

void	setParams (double alpha, double beta, double delta)
	Sets the parameters \(\alpha\), \(\beta\) and \(\delta\) of this object.

double []	getParams ()
	Return an array containing the parameters of the current object in regular order: [ \(\alpha\), \(\beta\), \(\delta\)].

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 alpha, double beta, double delta, double x)
	Computes and returns the density function.

static double	cdf (double alpha, double beta, double delta, double x)
	Computes and returns the distribution function.

static double	barF (double alpha, double beta, double delta, double x)
	Computes and returns the complementary distribution function \(1 - F(x)\).

static double	inverseF (double alpha, double beta, double delta, double u)
	Computes and returns the inverse distribution function.

static double []	getMLE (double[] x, int n, double delta)
	Given \(\delta=\) `delta`, estimates the parameters \((\alpha, \beta)\) of the Fréchet distribution using the maximum likelihood method with the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...

static FrechetDist	getInstanceFromMLE (double[] x, int n, double delta)
	Given \(\delta=\) `delta`, creates a new instance of a Fréchet distribution with parameters \(\alpha\) and \(\beta\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static double	getMean (double alpha, double beta, double delta)
	Returns the mean of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\). More...

static double	getVariance (double alpha, double beta, double delta)
	Returns the variance of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\). More...

static double	getStandardDeviation (double alpha, double beta, double delta)
	Returns the standard deviation of the Fréchet distribution with parameters \(\alpha\), \(\beta\) 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 Fréchet distribution [100] (page 3), with location parameter \(\delta\), scale parameter \(\beta> 0\), and shape parameter \(\alpha> 0\), where we use the notation \(z = (x-\delta)/\beta\).

It has density

\[ f (x) = \frac{\alpha e^{-z^{-\alpha}}}{\beta z^{\alpha+1}}, \qquad\mbox{for } x > \delta \]

and distribution function

\[ F(x) = e^{-z^{-\alpha}}, \qquad\mbox{for } x > \delta. \]

Both the density and the distribution are 0 for \(x \le\delta\).

The mean is given by

\[ E[X] = \delta+ \beta\Gamma\!\left(1 - \frac{1}{\alpha}\right), \]

where \(\Gamma(x)\) is the gamma function. The variance is

\[ \mbox{Var}[X] = \beta^2 \left[\Gamma\!\left(1 - \frac{2}{\alpha}\right) - \Gamma^2\!\left(1 - \frac{1}{\alpha}\right)\right]. \]

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 FrechetDist getInstanceFromMLE	(	double []	x,
		int	n,
		double	delta
	)

static

Given \(\delta=\) delta, creates a new instance of a Fréchet distribution with parameters \(\alpha\) and \(\beta\) 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
delta	location parameter

◆ getMean()

static double getMean	(	double	alpha,
		double	beta,
		double	delta
	)

static

Returns the mean of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\).

Returns: the mean

◆ getMLE()

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

static

Given \(\delta=\) delta, estimates the parameters \((\alpha, \beta)\) of the Fréchet distribution 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: [ \(\alpha\), \(\beta\)]. The maximum likelihood estimators are the values \((\hat{\alpha}, \hat{\beta})\) that satisfy the equations:

\begin{align*} \hat{\beta} & = \left(\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \delta)^{-\hat{\alpha}}\right)^{\!\!-1/\hat{\alpha}} \\ \frac{1}{n} \sum_{i=0}^{n-1} \ln(x_i - \delta) & = \frac{1}{\hat{\alpha}} + \frac{\sum_{i=0}^{n-1} (x_i - \delta)^{-\hat{\alpha}}\ln(x_i - \delta)}{\sum_{i=0}^{n-1} (x_i - \delta)^{-\hat{\alpha}}}. \end{align*}

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters
delta	location parameter

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

◆ getStandardDeviation()

static double getStandardDeviation	(	double	alpha,
		double	beta,
		double	delta
	)

static

Returns the standard deviation of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\).

Returns: the standard deviation

◆ getVariance()

static double getVariance	(	double	alpha,
		double	beta,
		double	delta
	)

static

Returns the variance of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\).

Returns: the variance

◆ 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:

FrechetDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()