Extends the class ContinuousDistribution for the Fréchet distribution [96] (page 3), with location parameter. More...

Inheritance diagram for umontreal.ssj.probdist.FrechetDist:

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)
	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	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 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 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 \(x[i]\), \(i = 0, 1,…, n-1\).
static FrechetDist	getInstanceFromMLE (double[] x, int n, double delta)
	Given \(\delta=\) delta, creates a new instance of a Fréchet distribution with parameters \(\alpha\) and.
static double	getMean (double alpha, double beta, double delta)
	Returns the mean of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\).
static double	getVariance (double alpha, double beta, double delta)
	Returns the variance of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\).
static double	getStandardDeviation (double alpha, double beta, double delta)
	Returns the standard deviation of the Fréchet distribution with parameters \(\alpha\), \(\beta\) and \(\delta\).

Detailed Description

Extends the class ContinuousDistribution for the Fréchet distribution [96] (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

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

Definition at line 52 of file FrechetDist.java.

Constructor & Destructor Documentation

◆ FrechetDist() [1/2]

umontreal.ssj.probdist.FrechetDist.FrechetDist ( double alpha )

Constructor for the standard Fréchet distribution with parameters \(\beta\) = 1 and \(\delta\) = 0.

Definition at line 149 of file FrechetDist.java.

◆ FrechetDist() [2/2]

umontreal.ssj.probdist.FrechetDist.FrechetDist	(	double	alpha,
		double	beta,
		double	delta )

Constructs a FrechetDist object with parameters \(\alpha\) = alpha, \(\beta\) = beta and \(\delta\) = delta.

Definition at line 157 of file FrechetDist.java.

Member Function Documentation

◆ barF() [1/2]

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

static

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

Definition at line 222 of file FrechetDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.FrechetDist.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 169 of file FrechetDist.java.

◆ cdf() [1/2]

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

static

Computes and returns the distribution function.

Definition at line 207 of file FrechetDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.FrechetDist.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 165 of file FrechetDist.java.

◆ density() [1/2]

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

static

Computes and returns the density function.

Definition at line 192 of file FrechetDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.FrechetDist.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 161 of file FrechetDist.java.

◆ getAlpha()

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

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

Definition at line 349 of file FrechetDist.java.

◆ getBeta()

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

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

Definition at line 356 of file FrechetDist.java.

◆ getDelta()

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

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

Definition at line 363 of file FrechetDist.java.

◆ getInstanceFromMLE()

FrechetDist umontreal.ssj.probdist.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

Definition at line 299 of file FrechetDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 177 of file FrechetDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.FrechetDist.getMean	(	double	alpha,
		double	beta,
		double	delta )

static

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

Returns: the mean

Definition at line 310 of file FrechetDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.FrechetDist.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 [ \(\hat{\alpha}\), \(\hat{\beta}\)]

Definition at line 275 of file FrechetDist.java.

◆ getParams()

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

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

Implements umontreal.ssj.probdist.Distribution.

Definition at line 385 of file FrechetDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 185 of file FrechetDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.FrechetDist.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

Definition at line 342 of file FrechetDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 181 of file FrechetDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.FrechetDist.getVariance	(	double	alpha,
		double	beta,
		double	delta )

static

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

Returns: the variance

Definition at line 325 of file FrechetDist.java.

◆ inverseF() [1/2]

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

static

Computes and returns the inverse distribution function.

Definition at line 237 of file FrechetDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.FrechetDist.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 173 of file FrechetDist.java.

◆ setParams()

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

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

Definition at line 371 of file FrechetDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 393 of file FrechetDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ FrechetDist() [1/2]

◆ FrechetDist() [2/2]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getAlpha()

◆ getBeta()

◆ getDelta()

◆ getInstanceFromMLE()

◆ 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()