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 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

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