# SuanShu, a Java numerical and statistical library

com.numericalmethod.suanshu.stats.evt.evd.univariate

## Class GeneralizedEVD

• java.lang.Object
• com.numericalmethod.suanshu.stats.evt.evd.univariate.GeneralizedEVD
• ### Constructor Summary

Constructors
Constructor and 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

All Methods
Modifier and Type Method and 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 Detail

• #### 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 Detail

• #### 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:
$$F(x)$$
See Also:
Wikipedia: Cumulative distribution function
• #### 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:
$$f(x)$$
See Also:
• #### 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)
See Also:
Wikipedia: Quantile function
• #### mean

public double mean()
Description copied from interface: ProbabilityDistribution
Gets the mean of this distribution.
Specified by:
mean in interface ProbabilityDistribution
Returns:
the mean
See Also:
Wikipedia: Expected value
• #### 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))
See Also:
Wikipedia: Moment-generating function
• #### skew

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

public double entropy()
Description copied from interface: ProbabilityDistribution
Gets the entropy of this distribution.
Specified by:
entropy in interface ProbabilityDistribution
Returns:
the entropy
See Also:
Wikipedia: Entropy (information theory)

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