hinverse-methods {vines}

Methods for the Inverse of the h-functions

Description

The h^{-1} function represents the inverse of the h function with respect to its first argument. It should be defined for every copula used in a pair-copula construction (or it will be evaluated numerically).

Usage

hinverse(copula, u, v, eps)

Arguments

copula

A bivariate copula object.

u

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 u, 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^{-1} function for a bivariate copula. This method is used if no particular definition is given for a copula. The inverse is calculated numerically using the uniroot function.

signature(copula = "indepCopula")

The h^{-1} function of the Independence copula.

h^{-1}(u, v) = u

signature(copula = "normalCopula")

The h^{-1} function of the normal copula.

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

signature(copula = "tCopula")

The h^{-1} function of the t copula.

h^{-1}(u, v; \rho, \nu) = t_{\nu} \left( t^{-1}_{\nu+1}(u)\ \sqrt{\frac{(\nu+(t^{-1}_{\nu}(v))^2)(1-\rho^2)}{\nu+1}} + \rho\ t^{-1}_{\nu}(v) \right)

signature(copula = "claytonCopula")

The h^{-1} function of the Clayton copula.

h^{-1}(u, v; \theta) = \left( \left( u\ v^{\theta+1}\right)^{-\frac{\theta}{\theta+1}} + 1 - v^{-\theta} \right)^{-1/\theta}

signature(copula = "frankCopula")

The h^{-1} function of the Frank copula.

h^{-1}(u, v; \theta) = -\log \left( 1 - \frac{1-e^{-\theta}} {(u^{-1} - 1) e^{-\theta v} + 1} \right) / \theta

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.