SSJ
V. 2.6.

umontreal.iro.lecuyer.probdist Class GumbelDist

```java.lang.Object
umontreal.iro.lecuyer.probdist.ContinuousDistribution
umontreal.iro.lecuyer.probdist.GumbelDist
```
All Implemented Interfaces:
Distribution

`public class GumbelDistextends 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
`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
` 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.
` 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 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.