Interface BivariateEVD

All Superinterfaces:
BivariateProbabilityDistribution, MultivariateProbabilityDistribution, RandomVectorGenerator, Seedable
All Known Implementing Classes:
AbstractBivariateEVD, BivariateEVDAsymmetricLogistic, BivariateEVDAsymmetricMixed, BivariateEVDAsymmetricNegativeLogistic, BivariateEVDBilogistic, BivariateEVDColesTawn, BivariateEVDHuslerReiss, BivariateEVDLogistic, BivariateEVDNegativeBilogistic, BivariateEVDNegativeLogistic

public interface BivariateEVD extends BivariateProbabilityDistribution, RandomVectorGenerator
Bivariate Extreme Value (BEV) distribution is the joint distribution of component-wise maxima of two-dimensional iid random vectors. Let \((\tilde{X}_i,\tilde{Y}_i)'\) be iid random vectors with joint distribution function \(\tilde{F}\). Also, let \(X_n=\max(\tilde{X}_1,\ldots,\tilde{X}_n)\) and \(Y_n=\max(\tilde{Y}_1,\ldots,\tilde{Y}_n)\), that is, \((X_n,Y_n)'\) is the vector of component-wise maxima. The limiting joint distribution function of normalized component-wise maxima, \[ G(x,y) = Pr(X_n \le x, Y_n \le y) \;\text{when}\; n \to \infty \] can be characterized in terms of the marginal GEV distributions and the BEV copula.
  • Method Details

    • spectralDensity

      double spectralDensity(double x)
      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.

      x - x
    • dependence

      double dependence(double x)
      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.

      x - x
    • conditionalCopula

      double conditionalCopula(double x1, double x2)
      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.
      x1 - an observation from \(U_1\)
      x2 - an observation from \(U_2\)
      the conditional copula \(P(U_1 < x_1|U_2 = x_2)\)