FrechetDist (Java Libraries for Stochastic Simulation)

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

Constructors
Constructor and Description
`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

Methods
Modifier and Type	Method and Description
`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.
`java.lang.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 java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class FrechetDist

Field Summary

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

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Constructor Detail

FrechetDist

FrechetDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMLE

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getAlpha

getBeta

getDelta

setParams

getParams

toString