BivariateEVDHuslerReiss (NMDEV 1.2.9 API)

java.lang.Object
- dev.nm.stat.distribution.multivariate.AbstractBivariateProbabilityDistribution
- - dev.nm.stat.evt.evd.bivariate.AbstractBivariateEVD
  - - dev.nm.stat.evt.evd.bivariate.BivariateEVDHuslerReiss

All Implemented Interfaces:

BivariateProbabilityDistribution, MultivariateProbabilityDistribution, BivariateEVD, RandomVectorGenerator, Seedable
```
public class BivariateEVDHuslerReiss
extends AbstractBivariateEVD
```
The Husler-Reiss model. Define \[ y_i = y_i(z_i) = \left[1+\frac{\xi_i(z_i-\mu_i)}{\sigma_i}\right]^{-1/\xi_i} \] for \(1+\xi_i(z_i-\mu_i)/\sigma_i > 0\) and \(i=1,2\) where the marginal univariate GEV parameters are given by \((\mu_i,\sigma_i,\xi_i)\), and \(G_i(z_i) = \exp(-y_i)\). The distribution function of the Husler-Reiss model is \[ G(z_1,z_2) = \exp\left(-y_1\Phi\left\{\frac{1}{r}+\frac{r}{2}\left[\log(\frac{y_1}{y_2})\right]\right\} -y_2\Phi\left\{\frac{1}{r}+\frac{r}{2}\left[\log(\frac{y_2}{y_1})\right]\right\}\right) \] where \(r > 0\) is the dependence parameter, and \(\Phi()\) is the standard normal distribution. Independence is obtained in the limit as \(r\) approaches zero. Complete dependence is obtained as \(r\) tends to infinity.
The R equivalent functions are evd::dbvhr, evd::pbvhr, evd::rbvhr, evd::hbvhr, evd::abvhr, evd::ccbvevd.

See Also:

"J. Husler and R.-D. Reiss, "Maxima of normal random vectors: between independence and complete dependence," in Statistics & Probability Letters, vol. 7, no. 4, 1989, pp. 283-286."

Constructor Summary

Constructors
Constructor and Description
`BivariateEVDHuslerReiss(double dependence)`
`BivariateEVDHuslerReiss(double dependence, GeneralizedEVD marginal)`
`BivariateEVDHuslerReiss(double dependence, GeneralizedEVD marginal1, GeneralizedEVD marginal2)`

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`cdf(double x1, double x2)` The joint distribution function \(F_{X_1,X_2}(x_1,x_2) = Pr(X_1 \le x_1, X_2 \le x_2)\).
`double`	`conditionalCopula(double x1, double x2)` The conditional copula function conditioning on either margin.
`double`	`density(double x1, double x2)` The joint distribution density \(f_{X_1,X_2}(x_1,x_2)\).
`double`	`dependence(double x)` The dependence function \(A\) for the parametric bivariate extreme value model.
`double[]`	`nextVector()` Gets the next random vector.
`void`	`seed(long... seeds)` Seed the random number/vector/scenario generator to produce repeatable experiments.
`double`	`spectralDensity(double x)` The density \(h\) of the spectral measure \(H\) on the interval (0,1).

Methods inherited from class dev.nm.stat.evt.evd.bivariate.AbstractBivariateEVD
covariance, entropy, mean, mode, moment

Methods inherited from class dev.nm.stat.distribution.multivariate.AbstractBivariateProbabilityDistribution
cdf, density

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface dev.nm.stat.distribution.multivariate.MultivariateProbabilityDistribution
cdf, density

- Constructor Detail
  - BivariateEVDHuslerReiss
```
public BivariateEVDHuslerReiss(double dependence)
```
  - BivariateEVDHuslerReiss
```
public BivariateEVDHuslerReiss(double dependence,
                               GeneralizedEVD marginal)
```
  - BivariateEVDHuslerReiss
```
public BivariateEVDHuslerReiss(double dependence,
                               GeneralizedEVD marginal1,
                               GeneralizedEVD marginal2)
```
- Method Detail
  - density
```
public double density(double x1,
                      double x2)
```
    Description copied from interface: BivariateProbabilityDistribution
    
    The joint distribution density \(f_{X_1,X_2}(x_1,x_2)\).
    
    Parameters:
    
    x1 - the value drawn from \(X_1\)
    
    x2 - the value drawn from \(X_2\)
    
    Returns:
    
    the joint density of \(X_1\) and \(X_2\)
  - cdf
```
public double cdf(double x1,
                  double x2)
```
    Description copied from interface: BivariateProbabilityDistribution
    
    The joint distribution function \(F_{X_1,X_2}(x_1,x_2) = Pr(X_1 \le x_1, X_2 \le x_2)\).
    
    Parameters:
    
    x1 - the value drawn from \(X_1\)
    
    x2 - the value drawn from \(X_2\)
    
    Returns:
    
    the joint distribution of \(X_1\) and \(X_2\)
  - spectralDensity
```
public double spectralDensity(double x)
```
    Description copied from interface: BivariateEVD
    
    The density \(h\) of the spectral measure \(H\) on the interval (0,1). Any bivariate extreme value distribution can be written as \[ G(z_1,z_2) = \exp\left\{-\int_0^1 \max(w y_1, (1-w) y_2) H(dw)\right\} \] where \(y_i=(1+\xi_i(z_i-\mu_i)/\sigma_i)^{(-1/\xi_i)}\), and \(\mu_i\), \(\sigma_i\), \(\xi_i\) are the location, scale and shape parameters.
    For some function \(H()\) defined on [0,1], satisfying \[ \int_0^1 w H(dw) = \int_0^1 (1-w) H(dw) = 1. \] \(H()\) is called the spectral measure, with density \(h\) on the interval (0,1).
    For differentiable models, \(H\) may have up to two point masses: at zero and one. Assuming that the model parameters are in the interior of the parameter space, we have the following. For the asymmetric logistic and asymmetric negative logistic models the point masses are of size \((1-t_1)\) and \((1-t_2)\) respectively. For the asymmetric mixed model they are of size \((1-\alpha-\beta)\) and \((1-\alpha-2*\beta)\) respectively. For all other models the point masses are zero.
    At independence, \(H\) has point masses of size one at both zero and one. At complete dependence [a non-differentiable model] \(H\) has a single point mass of size two at 1/2. In either case, \(h\) is zero everywhere.
    
    Parameters:
    
    x - x
    
    Returns:
    
    \(h(x)\)
  - dependence
```
public double dependence(double x)
```
    Description copied from interface: BivariateEVD
    
    The dependence function \(A\) for the parametric bivariate extreme value model. Any bivariate extreme value distribution can be written as \[ G(z_1,z_2) = \exp\left\{-(y_1+y_2)A\left[y_1/(y_1+y_2)\right]\right\} \] for some function \(A()\) defined on [0,1], where \(y_i=(1+\xi_i(z_i-\mu_i)/\sigma_i)^{(-1/\xi_i)}\), and \(\mu_i\), \(\sigma_i\), \(\xi_i\) are the location, scale and shape parameters.
    It follows that \(A(0)=A(1)=1\), and that \(A()\) is a convex function with \(\max(x,1-x) \le A(x) \le 1\) for all \(0 \le x \le 1\).
    The lower and upper limits of \(A\) are obtained under complete dependence and independence respectively. \(A()\) does not depend on the marginal parameters.
    
    Parameters:
    
    x - x
    
    Returns:
    
    \(A(x)\)
  - conditionalCopula
```
public double conditionalCopula(double x1,
                                double x2)
```
    Description copied from interface: BivariateEVD
    
    The conditional copula function conditioning on either margin. The function calculates \(P(U_1 < x_1|U_2 = x_2)\), where \((U_1,U_2)\) is a random vector with Uniform(0,1) margins and with a dependence structure given by the specified parametric model.
    
    Parameters:
    
    x1 - an observation from \(U_1\)
    
    x2 - an observation from \(U_2\)
    
    Returns:
    
    the conditional copula \(P(U_1 < x_1|U_2 = x_2)\)
    
    See Also:
    
    "A. G. Stephenson and J. A. Tawn, "Exploiting Occurence Times in Likelihood Inference for Component-wise Maxima," in Biometrika, vol. 92(1), 2004, pp. 213-217."
  - nextVector
```
public double[] nextVector()
```
    Description copied from interface: RandomVectorGenerator
    
    Gets the next random vector.
    
    Returns:
    
    the next random vector
  - seed
```
public void seed(long... seeds)
```
    Description copied from interface: Seedable
    
    Seed the random number/vector/scenario generator to produce repeatable experiments.
    
    Parameters:
    
    seeds - the seeds

Class BivariateEVDHuslerReiss

Constructor Summary

Method Summary

Methods inherited from class dev.nm.stat.evt.evd.bivariate.AbstractBivariateEVD

Methods inherited from class dev.nm.stat.distribution.multivariate.AbstractBivariateProbabilityDistribution

Methods inherited from class java.lang.Object

Methods inherited from interface dev.nm.stat.distribution.multivariate.MultivariateProbabilityDistribution

Constructor Detail

BivariateEVDHuslerReiss

BivariateEVDHuslerReiss

BivariateEVDHuslerReiss

Method Detail

density

cdf

spectralDensity

dependence

conditionalCopula

nextVector

seed