umontreal.iro.lecuyer.probdist
Class FrechetDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.FrechetDist

All Implemented Interfaces:

Distribution

public class FrechetDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Fréchet distribution, with location parameter δ, scale parameter β > 0, and shape parameter α > 0, where we use the notation z = (x - δ)/β. It has density

f (x) = αe^-z-α/(βz^α+1), for x > δ

and distribution function

F(x) = e^-z-α, for x > δ.

Both the density and the distribution are 0 for x <= δ.

The mean is given by

E[X] = δ + βΓ(1 - 1/α),

where Γ(x) is the gamma function. The variance is

Var[X] = β²[Γ(1 - 2/α) - (Γ(1 - 1/α))²].

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

FrechetDist(double alpha)
Constructor for the standard Fréchet distribution with parameters β = 1 and δ = 0.

FrechetDist(double alpha, double beta, double delta)
Constructs a FrechetDist object with parameters α = alpha, β = beta and δ = delta.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(double alpha, double beta, double delta, double x) Computes and returns the complementary distribution function 1 - F(x).
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double alpha, double beta, double delta, double x) Computes and returns the distribution function.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double alpha, double beta, double delta, double x) Computes and returns the density function.
double	getAlpha() Returns the parameter α of this object.
double	getBeta() Returns the parameter β of this object.
double	getDelta() Returns the parameter δ of this object.
static FrechetDist	getInstanceFromMLE(double[] x, int n, double delta) Given δ = delta, creates a new instance of a Fréchet distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
double	getMean() Returns the mean.
static double	getMean(double alpha, double beta, double delta) Returns the mean of the Fréchet distribution with parameters α, β and δ.
static double[]	getMLE(double[] x, int n, double delta) Given δ = delta, estimates the parameters (α, β) of the Fréchet distribution using the maximum likelihood method with the n observations x[i], i = 0, 1,…, n - 1.
double[]	getParams() Return an array containing the parameters of the current object in regular order: [α, β, δ].
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double alpha, double beta, double delta) Returns the standard deviation of the Fréchet distribution with parameters α, β and δ.
double	getVariance() Returns the variance.
static double	getVariance(double alpha, double beta, double delta) Returns the variance of the Fréchet distribution with parameters α, β and δ.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double alpha, double beta, double delta, double u) Computes and returns the inverse distribution function.
void	setParams(double alpha, double beta, double delta) Sets the parameters α, β and δ of this object.
String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

FrechetDist

public FrechetDist(double alpha)

Constructor for the standard Fréchet distribution with parameters β = 1 and δ = 0.

FrechetDist

public FrechetDist(double alpha,
                   double beta,
                   double delta)

Constructs a FrechetDist object with parameters α = alpha, β = beta and δ = delta.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class ContinuousDistribution

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

getMean

public double getMean()

Description copied from class: ContinuousDistribution

Returns the mean.

Specified by:

getMean in interface Distribution

Overrides:

getMean in class ContinuousDistribution

Returns:

the mean

getVariance

public double getVariance()

Description copied from class: ContinuousDistribution

Returns the variance.

Specified by:

getVariance in interface Distribution

Overrides:

getVariance in class ContinuousDistribution

Returns:

the variance

getStandardDeviation

public double getStandardDeviation()

Description copied from class: ContinuousDistribution

Returns the standard deviation.

Specified by:

getStandardDeviation in interface Distribution

Overrides:

getStandardDeviation in class ContinuousDistribution

Returns:

the standard deviation

density

public static double density(double alpha,
                             double beta,
                             double delta,
                             double x)

Computes and returns the density function.

cdf

public static double cdf(double alpha,
                         double beta,
                         double delta,
                         double x)

Computes and returns the distribution function.

barF

public static double barF(double alpha,
                          double beta,
                          double delta,
                          double x)

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

inverseF

public static double inverseF(double alpha,
                              double beta,
                              double delta,
                              double u)

Computes and returns the inverse distribution function.

getMLE

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

Given δ = delta, estimates the parameters (α, β) 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: [α, β].

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(α), hat(β)]

getInstanceFromMLE

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

Given δ = delta, creates a new instance of a Fréchet distribution with parameters α and β 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

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

Returns the mean of the Fréchet distribution with parameters α, β and δ.

Returns:

the mean

getVariance

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

Returns the variance of the Fréchet distribution with parameters α, β and δ.

Returns:

the variance

getStandardDeviation

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

Returns the standard deviation of the Fréchet distribution with parameters α, β and δ.

Returns:

the standard deviation

getAlpha

public double getAlpha()

Returns the parameter α of this object.

getBeta

public double getBeta()

Returns the parameter β of this object.

getDelta

public double getDelta()

Returns the parameter δ of this object.

setParams

public void setParams(double alpha,
                      double beta,
                      double delta)

Sets the parameters α, β and δ of this object.

getParams

public double[] getParams()

Return an array containing the parameters of the current object in regular order: [α, β, δ].

toString

public String toString()

Overrides:

toString in class Object