dev.nm.stat.evt.evd.univariate

Class MinimaDistribution

java.lang.Object

dev.nm.stat.evt.evd.univariate.MinimaDistribution

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

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 \(e^{tX}\).
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

MinimaDistribution

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:

\(f_{min}(x)\)

See Also:

• Wikipedia: Probability density function
• Wikipedia: Probability mass function

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:

\(F_{min}(x)\)

See Also:

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. \[ \begin{gathered} F^{-1}(u) = x, such that \\ Pr(X \le x) = u \end{gathered} \] 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 \(e^{tX}\). That is, \[ E(e^{tX}) \] 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)