SSJ
V. 2.6.

## 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/(βzα+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

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.