SSJ  3.3.1
Stochastic Simulation in Java
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ExtremeValueDist Class Reference

This class has been replaced by GumbelDist . More...

Inheritance diagram for ExtremeValueDist:
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Collaboration diagram for ExtremeValueDist:
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Public Member Functions

 ExtremeValueDist ()
 THIS CLASS HAS BEEN REPLACED BY GumbelDist . More...
 
 ExtremeValueDist (double alpha, double lambda)
 THIS CLASS HAS BEEN REPLACED BY GumbelDist . More...
 
double density (double x)
 
double cdf (double x)
 Returns the distribution function \(F(x)\). More...
 
double barF (double x)
 Returns \(\bar{F}(x) = 1 - F(x)\). More...
 
double inverseF (double u)
 Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More...
 
double getMean ()
 Returns the mean of the distribution function.
 
double getVariance ()
 Returns the variance of the distribution function.
 
double getStandardDeviation ()
 Returns the standard deviation of the distribution function.
 
double getAlpha ()
 Returns the parameter \(\alpha\) of this object.
 
double getLambda ()
 Returns the parameter \(\lambda\) of this object.
 
void setParams (double alpha, double lambda)
 Sets the parameters \(\alpha\) and \(\lambda\) of this object.
 
double [] getParams ()
 Return a table containing the parameters of the current distribution. More...
 
String toString ()
 Returns a String containing information about the current distribution.
 
- Public Member Functions inherited from ContinuousDistribution
abstract double density (double x)
 Returns \(f(x)\), the density evaluated at \(x\). More...
 
double barF (double x)
 Returns the complementary distribution function. More...
 
double inverseBrent (double a, double b, double u, double tol)
 Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...
 
double inverseBisection (double u)
 Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...
 
double inverseF (double u)
 Returns the inverse distribution function \(x = F^{-1}(u)\). More...
 
double getMean ()
 Returns the mean. More...
 
double getVariance ()
 Returns the variance. More...
 
double getStandardDeviation ()
 Returns the standard deviation. More...
 
double getXinf ()
 Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
 
double getXsup ()
 Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
 
void setXinf (double xa)
 Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
 
void setXsup (double xb)
 Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...
 

Static Public Member Functions

static double density (double alpha, double lambda, double x)
 Computes the density function.
 
static double cdf (double alpha, double lambda, double x)
 THIS CLASS HAS BEEN REPLACED BY GumbelDist . More...
 
static double barF (double alpha, double lambda, double x)
 Computes the complementary distribution function.
 
static double inverseF (double alpha, double lambda, double u)
 Computes the inverse distribution function.
 
static double [] getMLE (double[] x, int n)
 Estimates the parameters \((\alpha,\lambda)\) of the extreme value distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...
 
static double [] getMaximumLikelihoodEstimate (double[] x, int n)
 Same as getMLE.
 
static ExtremeValueDist getInstanceFromMLE (double[] x, int n)
 Creates a new instance of an extreme value distribution with parameters \(\alpha\) and \(\lambda\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...
 
static double getMean (double alpha, double lambda)
 Computes and returns the mean, \(E[X] = \alpha+ \gamma/\lambda\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\), where \(\gamma= 0.5772156649\) is the Euler-Mascheroni constant. More...
 
static double getVariance (double alpha, double lambda)
 Computes and returns the variance, \(\mbox{Var}[X] = \pi^2/(6\lambda^2)\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\). More...
 
static double getStandardDeviation (double alpha, double lambda)
 Computes and returns the standard deviation of the extreme value distribution with parameters \(\alpha\) and \(\lambda\). More...
 

Additional Inherited Members

- Public Attributes inherited from ContinuousDistribution
int decPrec = 15
 
- Protected Attributes inherited from ContinuousDistribution
double supportA = Double.NEGATIVE_INFINITY
 
double supportB = Double.POSITIVE_INFINITY
 
- Static Protected Attributes inherited from ContinuousDistribution
static final double XBIG = 100.0
 
static final double XBIGM = 1000.0
 
static final double [] EPSARRAY
 

Detailed Description

This class has been replaced by GumbelDist .

Extends the class ContinuousDistribution for the extreme value (or Gumbel) distribution [100]  (page 2), with location parameter \(\alpha\) and scale parameter \(\lambda> 0\). It has density

\[ f (x) = \lambda e^{-\lambda(x-\alpha)} e^{-e^{-\lambda(x-\alpha)}}, \qquad\qquad\mbox{for } -\infty< x < \infty, \tag{fextremevalue} \]

distribution function

\[ F(x) = e^{-e^{-\lambda(x - \alpha)}} \qquad\qquad\mbox{for } -\infty< x < \infty, \tag{Fextreme} \]

and inverse distribution function

\[ F^{-1}(u) = -\ln(-\ln(u))/\lambda+ \alpha, \qquad\mbox{for } 0 \le u \le1. \]

Constructor & Destructor Documentation

◆ ExtremeValueDist() [1/2]

THIS CLASS HAS BEEN REPLACED BY GumbelDist .

Constructs a ExtremeValueDist object with parameters \(\alpha\) = 0 and \(\lambda\) = 1.

◆ ExtremeValueDist() [2/2]

ExtremeValueDist ( double  alpha,
double  lambda 
)

THIS CLASS HAS BEEN REPLACED BY GumbelDist .

Constructs a ExtremeValueDist object with parameters \(\alpha\) = alpha and \(\lambda\) = lambda.

Member Function Documentation

◆ barF()

double barF ( double  x)

Returns \(\bar{F}(x) = 1 - F(x)\).

Parameters
xvalue at which the complementary distribution function is evaluated
Returns
complementary distribution function evaluated at x

Implements Distribution.

◆ cdf() [1/2]

double cdf ( double  x)

Returns the distribution function \(F(x)\).

Parameters
xvalue at which the distribution function is evaluated
Returns
distribution function evaluated at x

Implements Distribution.

◆ cdf() [2/2]

static double cdf ( double  alpha,
double  lambda,
double  x 
)
static

THIS CLASS HAS BEEN REPLACED BY GumbelDist .

Computes the distribution function.

◆ getInstanceFromMLE()

static ExtremeValueDist getInstanceFromMLE ( double []  x,
int  n 
)
static

Creates a new instance of an extreme value distribution with parameters \(\alpha\) and \(\lambda\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).

Parameters
xthe list of observations to use to evaluate parameters
nthe number of observations to use to evaluate parameters

◆ getMean()

static double getMean ( double  alpha,
double  lambda 
)
static

Computes and returns the mean, \(E[X] = \alpha+ \gamma/\lambda\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\), where \(\gamma= 0.5772156649\) is the Euler-Mascheroni constant.

Returns
the mean of the Extreme Value distribution \(E[X] = \alpha+ \gamma/ \lambda\)

◆ getMLE()

static double [] getMLE ( double []  x,
int  n 
)
static

Estimates the parameters \((\alpha,\lambda)\) of the extreme value distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).

The estimates are returned in a two-element array, in regular order: [ \(\alpha\), \(\lambda\)]. The maximum likelihood estimators are the values \((\hat{\alpha}, \hat{\lambda})\) that satisfy the equations:

\begin{align*} \hat{\lambda} & = \bar{x}_n - \frac{\sum_{i=1}^n x_i  e^{- \hat{\lambda} x_i}}{\sum_{i=1}^n e^{-\hat{\lambda} x_i}} \\ \hat{\alpha} & = - \frac{1}{\hat{\lambda}} \ln\left( \frac{1}{n} \sum_{i=1}^n e^{-\hat{\lambda} x_i} \right), \end{align*}

where \(\bar{x}_n\) is the average of \(x[0],…,x[n-1]\) [57]  (page 89).

Parameters
xthe list of observations used to evaluate parameters
nthe number of observations used to evaluate parameters
Returns
returns the parameters [ \(\hat{\alpha}\), \(\hat{\lambda}\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha\), \(\lambda\)].

Implements Distribution.

◆ getStandardDeviation()

static double getStandardDeviation ( double  alpha,
double  lambda 
)
static

Computes and returns the standard deviation of the extreme value distribution with parameters \(\alpha\) and \(\lambda\).

Returns
the standard deviation of the extreme value distribution

◆ getVariance()

static double getVariance ( double  alpha,
double  lambda 
)
static

Computes and returns the variance, \(\mbox{Var}[X] = \pi^2/(6\lambda^2)\), of the extreme value distribution with parameters \(\alpha\) and \(\lambda\).

Returns
the variance of the extreme value distribution \(\mbox{Var}[X] = 1/6 \pi^2 1/\lambda^2\)

◆ inverseF()

double inverseF ( double  u)

Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).

Parameters
uvalue in the interval \((0,1)\) for which the inverse distribution function is evaluated
Returns
the inverse distribution function evaluated at u

Implements Distribution.


The documentation for this class was generated from the following file: