h-functions and inverse h-functions

Vine algorithms are built from conditional distribution functions of bivariate copulas. For a bivariate copula with arguments , VineCopulas.jl uses the convention

The public API is:

hfunc1(C, u, v)
hfunc2(C, u, v)
hinv1(C, q, v)
hinv2(C, q, u)

hinv1 solves for the first coordinate given and . hinv2 solves for the second coordinate given and .

Example

using VineCopulas

C = ClaytonCopula(2, 2.0)
u, v = 0.25, 0.75

q1 = hfunc1(C, u, v)
q2 = hfunc2(C, u, v)

(u, hinv1(C, q1, v), v, hinv2(C, q2, u))

(0.25, 0.24999999999999997, 0.75, 0.7499999999999991)

Why inverses matter

Simulation from a vine is recursive. The algorithm starts with independent uniforms and then applies inverse conditional distributions. Therefore, every pair-copula used for simulation must have reliable hinv1 and hinv2 methods.

Density evaluation only needs the pair density and the forward h-functions, but Rosenblatt inversion and random generation require the inverse h-functions.

Generic and specialized paths

The package provides a generic fallback based on automatic differentiation and safeguarded scalar root-finding. Specialized methods are implemented for families where analytic formulas or stable coordinate systems are available.

This design makes contribution straightforward: a new bivariate copula can first rely on generic fallbacks; if tests expose instability or performance issues, family-specific hfunc/hinv methods can be added.