# SuanShu, a Java numerical and statistical library

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

• java.lang.Object
• All Implemented Interfaces:
ProbabilityDistribution, UnivariateEVD

public class MinimaDistribution
extends Object
implements UnivariateEVD
The distribution of $$M$$, where $$M=\min(x_1,x_2,...,x_n)$$ and $$x_i$$'s are iid samples drawn from of a random variable $$X$$ with cdf $$F(x)$$. Since $$x_i$$'s are independent, the cumulative distribution function of $$M$$ is simply $\begin{eqnarray} F_{min}(x;n) & = & Pr(\min(X_1,X_2,...,X_n) \le x) \\ & = & 1-Pr(\min(X_1,X_2,...,X_n) > x) \\ & = & 1-Pr(\text{all} \; X_i > x) \\ & = & 1-\prod_{i=1}^n(1-F(x)) \\ & = & 1-[1-F(x)]^n \end{eqnarray}$

The R equivalent functions are evd::dextreme, evd::pextreme, evd::qextreme, evd::rextreme.

• ### Constructor Summary

Constructors
Constructor and Description
MinimaDistribution(ProbabilityDistribution dist, int nIIDs)
• ### Method Summary

All Methods
Modifier and Type Method and Description
double cdf(double x)
The cumulative distribution function.
double density(double x)
The probability density function.
double entropy()
Gets the entropy of this distribution.
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 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

public MinimaDistribution(ProbabilityDistribution dist,
int nIIDs)
• ### Method Detail

• #### density

public double density(double x)
The probability density function. That is, $f_{min}(x;n) = \frac{d}{dx}F_{min}(x;n) = nf(x)[1-F(x)]^{n-1}.$
Specified by:
density in interface ProbabilityDistribution
Parameters:
x - $$x$$
Returns:
$$f_{min}(x)$$
• #### 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:
$$\log(f(x))$$
• #### cdf

public double cdf(double x)
The cumulative distribution function. That is, $f_{min}(x;n) = 1-[1-F(x)]^n.$
Specified by:
cdf in interface ProbabilityDistribution
Parameters:
x - $$x$$
Returns:
$$F_{min}(x)$$
Wikipedia: Cumulative distribution function
• #### 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)
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
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))
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
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
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
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
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