Package dev.nm.stat.evt.evd.univariate
Class MinimaDistribution
- java.lang.Object
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- dev.nm.stat.evt.evd.univariate.MinimaDistribution
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- 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 areevd::dextreme,evd::pextreme,evd::qextreme,evd::rextreme.
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Constructor Summary
Constructors Constructor Description MinimaDistribution(ProbabilityDistribution dist, int nIIDs)
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Method Summary
All Methods Instance Methods Concrete Methods Modifier and Type Method Description doublecdf(double x)The cumulative distribution function.doubledensity(double x)The probability density function.doubleentropy()Gets the entropy of this distribution.doublekurtosis()Gets the excess kurtosis of this distribution.doublelogDensity(double x)Get the logarithm of the probability density function at \(x\), that is, \(\log(f(x))\).doublemean()Gets the mean of this distribution.doublemedian()Gets the median of this distribution.doublemoment(double x)The moment generating function is the expected value of etX.doublequantile(double p)Gets the quantile, the inverse of the cumulative distribution function.doubleskew()Gets the skewness of this distribution.doublevariance()Gets the variance of this distribution.
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Constructor Detail
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MinimaDistribution
public MinimaDistribution(ProbabilityDistribution dist, int nIIDs)
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Method Detail
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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:
densityin interfaceProbabilityDistribution- Parameters:
x- \(x\)- Returns:
- \(f_{min}(x)\)
- See Also:
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logDensity
public double logDensity(double x)
Description copied from interface:UnivariateEVDGet the logarithm of the probability density function at \(x\), that is, \(\log(f(x))\).- Specified by:
logDensityin interfaceUnivariateEVD- Parameters:
x- \(x\)- Returns:
- \(\log(f(x))\)
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cdf
public double cdf(double x)
The cumulative distribution function. That is, \[ f_{min}(x;n) = 1-[1-F(x)]^n. \]- Specified by:
cdfin interfaceProbabilityDistribution- Parameters:
x- \(x\)- Returns:
- \(F_{min}(x)\)
- See Also:
- Wikipedia: Cumulative distribution function
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quantile
public double quantile(double p)
Description copied from interface:ProbabilityDistributionGets 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.
This may not always exist.F-1(u) = x, such that Pr(X ≤ x) = u
- Specified by:
quantilein interfaceProbabilityDistribution- Parameters:
p-u, a quantile- Returns:
- F-1(u)
- See Also:
- Wikipedia: Quantile function
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mean
public double mean()
Description copied from interface:ProbabilityDistributionGets the mean of this distribution.- Specified by:
meanin interfaceProbabilityDistribution- Returns:
- the mean
- See Also:
- Wikipedia: Expected value
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moment
public double moment(double x)
Description copied from interface:ProbabilityDistributionThe moment generating function is the expected value of etX. That is,E(etX)
This may not always exist.- Specified by:
momentin interfaceProbabilityDistribution- Parameters:
x- t- Returns:
- E(exp(tX))
- See Also:
- Wikipedia: Moment-generating function
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skew
public double skew()
Description copied from interface:ProbabilityDistributionGets the skewness of this distribution.- Specified by:
skewin interfaceProbabilityDistribution- Returns:
- the skewness
- See Also:
- Wikipedia: Skewness
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variance
public double variance()
Description copied from interface:ProbabilityDistributionGets the variance of this distribution.- Specified by:
variancein interfaceProbabilityDistribution- Returns:
- the variance
- See Also:
- Wikipedia: Variance
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median
public double median()
Description copied from interface:ProbabilityDistributionGets the median of this distribution.- Specified by:
medianin interfaceProbabilityDistribution- Returns:
- the median
- See Also:
- Wikipedia: Median
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kurtosis
public double kurtosis()
Description copied from interface:ProbabilityDistributionGets the excess kurtosis of this distribution.- Specified by:
kurtosisin interfaceProbabilityDistribution- Returns:
- the excess kurtosis
- See Also:
- Wikipedia: Kurtosis
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entropy
public double entropy()
Description copied from interface:ProbabilityDistributionGets the entropy of this distribution.- Specified by:
entropyin interfaceProbabilityDistribution- Returns:
- the entropy
- See Also:
- Wikipedia: Entropy (information theory)
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