# Class GeneralizedEVD

java.lang.Object
dev.nm.stat.evt.evd.univariate.GeneralizedEVD
All Implemented Interfaces:
ProbabilityDistribution, UnivariateEVD
Direct Known Subclasses:
FrechetDistribution, GumbelDistribution, ReversedWeibullDistribution

public class GeneralizedEVD extends Object implements UnivariateEVD
Generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the limit distribution of properly normalized maxima of a sequence of independent and identically distributed (IID) random variables.

The R equivalent functions are evd::dgev, evd::pgev, evd::qgev, evd::mtransform.

• ## Constructor Summary

Constructors
Constructor
Description
GeneralizedEVD()
Create an instance of generalized extreme value distribution with the default parameter values: location $$\mu=0$$, scale $$\sigma=1$$, shape $$\xi=0$$.
GeneralizedEVD(double location, double scale, double shape)
Create an instance of generalized extreme value distribution with the given parameters.
• ## Method Summary

Modifier and Type
Method
Description
double
cdf(double x)
Gets the cumulative probability F(x) = Pr(X ≤ x).
double
density(double x)
The density function, which, if exists, is the derivative of F.
double
entropy()
Gets the entropy of this distribution.
double
getLocation()
Get the location parameter.
double
getScale()
Get the scale parameter.
double
getShape()
Get the shape parameter.
double
kurtosis()
Gets the excess kurtosis of this distribution.
double
logDensity(double x)
Get the logarithm of the probability density function at $$x$$, that is, $$\log(f(x))$$.
double
marginalInverseTransform(double x)
Inverse of marginal transform.
double
marginalTransform(double x)
Transform to exponential margins under the GEV model.
double
mean()
Gets the mean of this distribution.
double
median()
Gets the median of this distribution.
double
moment(double x)
The moment generating function is the expected value of etX.
double
quantile(double p)
Gets the quantile, the inverse of the cumulative distribution function.
double
skew()
Gets the skewness of this distribution.
double
variance()
Gets the variance of this distribution.

### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ## Constructor Details

• ### GeneralizedEVD

public GeneralizedEVD()
Create an instance of generalized extreme value distribution with the default parameter values: location $$\mu=0$$, scale $$\sigma=1$$, shape $$\xi=0$$.
• ### GeneralizedEVD

public GeneralizedEVD(double location, double scale, double shape)
Create an instance of generalized extreme value distribution with the given parameters.
Parameters:
location - the location parameter $$\mu$$
scale - the scale parameter $$\sigma > 0$$
shape - the shape parameter $$\xi$$
• ## Method Details

• ### getLocation

public double getLocation()
Get the location parameter.
Returns:
$$\mu$$
• ### getScale

public double getScale()
Get the scale parameter.
Returns:
$$\sigma$$
• ### getShape

public double getShape()
Get the shape parameter.
Returns:
$$\xi$$
• ### marginalTransform

public double marginalTransform(double x)
Transform to exponential margins under the GEV model. That is, /[ t(x) = \begin{cases} \big(1+(\tfrac{x-\mu}{\sigma})\xi\big)^{-1/\xi} \textrm{if}\ \xi\neq0 \\ e^{-(x-\mu)/\sigma} \textrm{if}\ \xi=0 \end{cases} /]
Parameters:
x - $$x$$
Returns:
$$t(x)$$
• ### marginalInverseTransform

public double marginalInverseTransform(double x)
Inverse of marginal transform.
Parameters:
x - $$x$$
Returns:
$$t^{-1}(x)$$
• ### cdf

public double cdf(double x)
Gets the cumulative probability F(x) = Pr(X ≤ x). The cumulative distribution function of GEV distribution is $F(x;\mu,\sigma,\xi) = \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}$ for $$1+\xi(x-\mu)/\sigma>0$$.
Specified by:
cdf in interface ProbabilityDistribution
Parameters:
x - $$x$$
Returns:
$$F(x)$$
• ### density

public double density(double x)
The density function, which, if exists, is the derivative of F. It describes the density of probability at each point in the sample space.
f(x) = dF(X) / dx
This may not always exist.

For the discrete cases, this is the probability mass function. It gives the probability that a discrete random variable is exactly equal to some value. The probability density function of GEV distribution is $f(x;\mu,\sigma,\xi) = \frac{1}{\sigma}\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{(-1/\xi)-1} \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}$ for $$1+\xi(x-\mu)/\sigma>0$$.

Specified by:
density in interface ProbabilityDistribution
Parameters:
x - $$x$$
Returns:
$$f(x)$$
• ### logDensity

public double logDensity(double x)
Description copied from interface: UnivariateEVD
Get the logarithm of the probability density function at $$x$$, that is, $$\log(f(x))$$.
Specified by:
logDensity in interface UnivariateEVD
Parameters:
x - $$x$$
Returns:
$$\log(f(x))$$
• ### quantile

public double quantile(double p)
Description copied from interface: ProbabilityDistribution
Gets the quantile, the inverse of the cumulative distribution function. It is the value below which random draws from the distribution would fall u×100 percent of the time.

F-1(u) = x, such that
Pr(X ≤ x) = u

This may not always exist.
Specified by:
quantile in interface ProbabilityDistribution
Parameters:
p - u, a quantile
Returns:
F-1(u)
• ### mean

public double mean()
Description copied from interface: ProbabilityDistribution
Gets the mean of this distribution.
Specified by:
mean in interface ProbabilityDistribution
Returns:
the mean
• ### moment

public double moment(double x)
Description copied from interface: ProbabilityDistribution
The moment generating function is the expected value of etX. That is,
E(etX)
This may not always exist.
Specified by:
moment in interface ProbabilityDistribution
Parameters:
x - t
Returns:
E(exp(tX))
• ### skew

public double skew()
Description copied from interface: ProbabilityDistribution
Gets the skewness of this distribution.
Specified by:
skew in interface ProbabilityDistribution
Returns:
the skewness
• ### variance

public double variance()
Description copied from interface: ProbabilityDistribution
Gets the variance of this distribution.
Specified by:
variance in interface ProbabilityDistribution
Returns:
the variance
• ### median

public double median()
Description copied from interface: ProbabilityDistribution
Gets the median of this distribution.
Specified by:
median in interface ProbabilityDistribution
Returns:
the median
• ### kurtosis

public double kurtosis()
Description copied from interface: ProbabilityDistribution
Gets the excess kurtosis of this distribution.
Specified by:
kurtosis in interface ProbabilityDistribution
Returns:
the excess kurtosis
Description copied from interface: ProbabilityDistribution
entropy in interface ProbabilityDistribution