h-methods function

Methods for the h-functions

Methods for the h-functions

The hh function represents the conditional distribution function of a bivariate copula and it should be defined for every copula used in a pair-copula construction. It is defined as the partial derivative of the distribution function of the copula w.r.t. the second argument h(x,v)=F(x∣v)=∂C(x,v)/∂vh(x,v) = F(x|v) = \partial C(x,v) / \partial v. methods

h(copula, x, v, eps)

Arguments

  • copula: A bivariate copula object.
  • x: Numeric vector with values in [0,1][0,1].
  • v: Numeric vector with values in [0,1][0,1].
  • eps: To avoid numerical problems for extreme values, the values of x, v and return values close to 0 and 1 are substituted by eps and 1 - eps, respectively. The default eps value for most of the copulas is .Machine$double.eps^0.5.

Methods

  • signature(copula = "copula"): Default definition of the hh function for a bivariate copula. This method is used if no particular definition is given for a copula. The partial derivative is calculated numerically using the numericDeriv function.
  • signature(copula = "indepCopula"): The hh function of the independence copula.
h(x,v)=x h(x, v) = x
  • signature(copula = "normalCopula"): The hh function of the normal copula.
h(x,v;ρ)=Φ(Φ−1(x)−ρ Φ−1(v)1−ρ2) h(x, v; \rho) =\Phi \left( \frac{\Phi^{-1}(x) - \rho\ \Phi^{-1}(v)}{\sqrt{1-\rho^2}} \right)
  • signature(copula = "tCopula"): The hh function of the t copula.
h(x,v;ρ,ν)=tν+1(tν−1(x)−ρ tν−1(v)(ν+(tν−1(v))2)(1−ρ2)ν+1) h(x, v; \rho, \nu) =t_{\nu+1} \left( \frac{t^{-1}_{\nu}(x) - \rho\ t^{-1}_{\nu}(v)}{\sqrt{\frac{(\nu+(t^{-1}_{\nu}(v))^2)(1-\rho^2)}{\nu+1}}} \right)
  • signature(copula = "claytonCopula"): The hh function of the Clayton copula.
h(x,v;θ)=v−θ−1(x−θ+v−θ−1)−1−1/θ h(x, v; \theta) = v^{-\theta-1}(x^{-\theta}+v^{-\theta}-1)^{-1-1/\theta}
  • signature(copula = "gumbelCopula"): The hh function of the Gumbel copula.
h(x,v;θ)=C(x,v;θ) 1v (−log⁡v)θ−1((−log⁡x)θ+(−log⁡v)θ)1/θ−1 h(x, v; \theta) = C(x, v; \theta)\ \frac{1}{v}\ (-\log v)^{\theta-1}\left((-\log x)^{\theta} + (-\log v)^{\theta} \right)^{1/\theta-1}
  • signature(copula = "fgmCopula"): The hh function of the Farlie-Gumbel-Morgenstern copula.
h(x,v;θ)=(1+θ (−1+2v) (−1+x)) x h(x, v; \theta) =(1 + \theta \ (-1 + 2v) \ (-1 + x)) \ x
  • signature(copula = "frankCopula"): The hh function of the Frank copula.
h(x,v;θ)=e−θv1−e−θ1−e−θx+e−θv−1 h(x, v; \theta) =\frac{e^{-\theta v}}{\frac{1 - e^{-\theta}}{1 - e^{-\theta x}} + e^{-\theta v} - 1}
  • signature(copula = "galambosCopula"): The hh function of the Galambos copula.
h(x,v;θ)=C(x,v;θ)v(1−[1+(−log⁡v−log⁡x)θ]−1−1/θ) h(x, v; \theta) =\frac{C(x, v; \theta)}{v}\left( 1 - \left[ 1 + \left(\frac{-\log v}{-\log x}\right)^{\theta} \right]^{-1-1/\theta} \right)

References

Aas, K. and Czado, C. and Frigessi, A. and Bakken, H. (2009) Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics 44 , 182--198.

Schirmacher, D. and Schirmacher, E. (2008) Multivariate dependence modeling using pair-copulas. Enterprise Risk Management Symposium, Chicago.

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