h-methods {vines}

Methods for the h-functions

Description

The h 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) = \partial C(x,v) / \partial v.

Usage

h(copula, x, v, eps)

Arguments

copula

A bivariate copula object.

x

Numeric vector with values in [0,1].

v

Numeric vector with values in [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 h 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 h function of the independence copula.

h(x, v) = x

signature(copula = "normalCopula")

The h function of the normal copula.

h(x, v; \rho) = \Phi \left( \frac{\Phi^{-1}(x) - \rho\ \Phi^{-1}(v)} {\sqrt{1-\rho^2}} \right)

signature(copula = "tCopula")

The h function of the t copula.

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 h function of the Clayton copula.

h(x, v; \theta) = v^{-\theta-1}(x^{-\theta}+v^{-\theta}-1)^{-1-1/\theta}

signature(copula = "gumbelCopula")

The h function of the Gumbel copula.

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 h function of the Farlie-Gumbel-Morgenstern copula.

h(x, v; \theta) = (1 + \theta \ (-1 + 2v) \ (-1 + x)) \ x

signature(copula = "frankCopula")

The h function of the Frank copula.

h(x, v; \theta) = \frac{e^{-\theta v}} {\frac{1 - e^{-\theta}}{1 - e^{-\theta x}} + e^{-\theta v} - 1}

signature(copula = "galambosCopula")

The h function of the Galambos copula.

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.