GumbelDist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.ContinuousDistribution
- - umontreal.iro.lecuyer.probdist.GumbelDist

All Implemented Interfaces:

Distribution
```
public class GumbelDist
extends ContinuousDistribution
```
Extends the class ContinuousDistribution for the Gumbel distribution, with location parameter δ and scale parameter β≠ 0. Using the notation z = (x - δ)/β, it has density

f (x) = e^-ze^{-e^-z}/| β|,        for - ∞ < x < ∞.
and distribution function

F(x) = e^{-e^-z},        for β > 0

F(x) = 1 - e^{-e^-z},        for β < 0.

Field Summary
- Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
  decPrec

Constructor Summary

Constructors
Constructor and Description
`GumbelDist()` Constructor for the standard Gumbel distribution with parameters β = 1 and δ = 0.
`GumbelDist(double beta, double delta)` Constructs a `GumbelDist` object with parameters 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 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 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 beta, double delta, double x)` Computes and returns the density function.
`double`	`getBeta()` Returns the parameter β of this object.
`double`	`getDelta()` Returns the parameter δ of this object.
`static GumbelDist`	`getInstanceFromMLE(double[] x, int n)` Creates a new instance of an Gumbel distribution with parameters β and δ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1, assuming that β > 0.
`static GumbelDist`	`getInstanceFromMLEmin(double[] x, int n)` Similar to `getInstanceFromMLE`, but for the case β < 0.
`double`	`getMean()` Returns the mean.
`static double`	`getMean(double beta, double delta)` Returns the mean, E[X] = δ + γβ, of the Gumbel distribution with parameters β and δ, where γ = 0.5772156649015329 is the Euler-Mascheroni constant.
`static double[]`	`getMLE(double[] x, int n)` Estimates the parameters (β, δ) of the Gumbel distribution, assuming that β > 0, and using the maximum likelihood method with the n observations x[i], i = 0, 1,…, n - 1.
`static double[]`	`getMLEmin(double[] x, int n)` Similar to `getMLE`, but for the case β < 0.
`double[]`	`getParams()` Return a table containing the parameters of the current distribution.
`double`	`getStandardDeviation()` Returns the standard deviation.
`static double`	`getStandardDeviation(double beta, double delta)` Returns the standard deviation of the Gumbel distribution with parameters β and δ.
`double`	`getVariance()` Returns the variance.
`static double`	`getVariance(double beta, double delta)` Returns the variance Var[X] = π²β²/6 of the Gumbel distribution with parameters β and δ.
`double`	`inverseF(double u)` Returns the inverse distribution function x = F^-1(u).
`static double`	`inverseF(double beta, double delta, double u)` Computes and returns the inverse distribution function.
`void`	`setParams(double beta, double delta)` Sets the parameters 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
  - GumbelDist
```
public GumbelDist()
```
    Constructor for the standard Gumbel distribution with parameters β = 1 and δ = 0.
  - GumbelDist
```
public GumbelDist(double beta,
          double delta)
```
    Constructs a GumbelDist object with parameters 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 beta,
             double delta,
             double x)
```
    Computes and returns the density function.
  - cdf
```
public static double cdf(double beta,
         double delta,
         double x)
```
    Computes and returns the distribution function.
  - barF
```
public static double barF(double beta,
          double delta,
          double x)
```
    Computes and returns the complementary distribution function 1 - F(x).
  - inverseF
```
public static double inverseF(double beta,
              double delta,
              double u)
```
    Computes and returns the inverse distribution function.
  - getMLE
```
public static double[] getMLE(double[] x,
              int n)
```
    Estimates the parameters (β, δ) of the Gumbel distribution, assuming that β > 0, and 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: [β, δ].
    
    Parameters:
    x - the list of observations used to evaluate parameters
    n - the number of observations used to evaluate parameters
    
    Returns:
    returns the parameters [ hat(δ), hat(β)]
  - getMLEmin
```
public static double[] getMLEmin(double[] x,
                 int n)
```
    Similar to getMLE, but for the case β < 0.
    
    Parameters:
    x - the list of observations used to evaluate parameters
    n - the number of observations used to evaluate parameters
    
    Returns:
    returns the parameters [ hat(δ), hat(β)]
  - getInstanceFromMLE
```
public static GumbelDist getInstanceFromMLE(double[] x,
                            int n)
```
    Creates a new instance of an Gumbel distribution with parameters β and δ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1, assuming that β > 0.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
  - getInstanceFromMLEmin
```
public static GumbelDist getInstanceFromMLEmin(double[] x,
                               int n)
```
    Similar to getInstanceFromMLE, but for the case β < 0.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
  - getMean
```
public static double getMean(double beta,
             double delta)
```
    Returns the mean, E[X] = δ + γβ, of the Gumbel distribution with parameters β and δ, where γ = 0.5772156649015329 is the Euler-Mascheroni constant.
    
    Returns:
    the mean of the Extreme Value distribution E[X] = δ + γ*β
  - getVariance
```
public static double getVariance(double beta,
                 double delta)
```
    Returns the variance Var[X] = π²β²/6 of the Gumbel distribution with parameters β and δ.
    
    Returns:
    the variance of the Gumbel distribution Var[X] = ()πβ)²/6
  - getStandardDeviation
```
public static double getStandardDeviation(double beta,
                          double delta)
```
    Returns the standard deviation of the Gumbel distribution with parameters β and δ.
    
    Returns:
    the standard deviation of the Gumbel distribution
  - 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 beta,
             double delta)
```
    Sets the parameters β and δ of this object.
  - getParams
```
public double[] getParams()
```
    Return a table containing the parameters of the current distribution. This table is put in regular order: [β, δ].
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class GumbelDist

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

GumbelDist

GumbelDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMLE

getMLEmin

getInstanceFromMLE

getInstanceFromMLEmin

getMean

getVariance

getStandardDeviation

getBeta

getDelta

setParams

getParams

toString