OrderStatisticsDistribution (NMDEV 1.2.9 API)

java.lang.Object
- dev.nm.stat.evt.evd.univariate.OrderStatisticsDistribution

All Implemented Interfaces:

ProbabilityDistribution, UnivariateEVD
```
public class OrderStatisticsDistribution
extends Object
implements UnivariateEVD
```
The asymptotic nondegenerate distributions of the r-th smallest (largest) order statistic.
The R equivalent functions are evd::dorder, evd::porder, evd::rorder.

See Also:

"E. Castillo, A. S. Hadi, N. Balakrishnan, J. M. Sarabia, "Extreme Value and Related Models with Applications in Engineering and Science," Wiley-Interscience, 2004, ch.7, P.158"

Constructor Summary

Constructors
Constructor and Description
`OrderStatisticsDistribution(ProbabilityDistribution dist, int nIIDs, int order)` Create an instance with the probability distribution of \(X\), the number of iid samples to be drawn, and the order statistic.

Method Summary

All Methods Instance Methods Concrete 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`	`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 x)` 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
  - OrderStatisticsDistribution
```
public OrderStatisticsDistribution(ProbabilityDistribution dist,
                                   int nIIDs,
                                   int order)
```
    Create an instance with the probability distribution of \(X\), the number of iid samples to be drawn, and the order statistic.
    
    Parameters:
    
    dist - the probability distribution
    
    nIIDs - the number of independent variables
    
    order - the order statistic (largest) (1 means smallest, n means largest)
- Method Detail
  - density
```
public double density(double x)
```
    Description copied from interface: ProbabilityDistribution
    
    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.
    Specified by:
    
    density in interface ProbabilityDistribution
    
    Parameters:
    
    x - x
    
    Returns:
    
    f(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)
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the cumulative probability F(x) = Pr(X ≤ x).
    
    Specified by:
    
    cdf in interface ProbabilityDistribution
    
    Parameters:
    
    x - x
    
    Returns:
    
    F(x) = Pr(X ≤ x)
    
    See Also:
    
    Wikipedia: Cumulative distribution function
  - quantile
```
public double quantile(double x)
```
    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:
    
    x - 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)

Class OrderStatisticsDistribution

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

OrderStatisticsDistribution

Method Detail

density

logDensity

cdf

quantile

mean

moment

skew

variance

median

kurtosis

entropy