DensityGenerator.normalize: Normalization of an elliptical copula generator

In ElliptCopulas: Inference of Elliptical Distributions and Copulas

Normalization of an elliptical copula generator

Description

The function DensityGenerator.normalize transforms an elliptical copula generator into an elliptical copula generator,generating the same distribution and which is normalized to follow the normalization constraint

\frac{\pi^{d/2}}{\Gamma(d/2)} \int_0^{+\infty} g_k(t) t^{(d-2)/2} dt = 1.

as well as the identification constraint

\frac{\pi^{(d-1)/2}}{\Gamma((d-1)/2)} \int_0^{+\infty} g_k(t) t^{(d-3)/2} dt = b.

The function DensityGenerator.check checks, for a given generator, whether these two constraints are satisfied.

Usage

DensityGenerator.normalize(grid, grid_g, d, verbose = 0, b = 1)

DensityGenerator.check(grid, grid_g, d, b = 1)

Arguments

grid	the regularly spaced grid on which the values of the generator are given.
grid_g	the values of the d-dimensional generator at points of the grid.
d	the dimension of the space.
verbose	if 1, prints the estimated (alpha, beta) such that new_g(t) = alpha * old_g(beta*t).
b	the target value for the identification constraint.

Value

DensityGenerator.normalize returns the normalized generator, as a list of values on the same grid.

DensityGenerator.check returns (invisibly) a vector of two booleans where the first element is TRUE if the normalization constraint is satisfied and the second element is TRUE if the identification constraint is satisfied.

References

Derumigny, A., & Fermanian, J. D. (2022). Identifiability and estimation of meta-elliptical copula generators. Journal of Multivariate Analysis, article 104962. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jmva.2022.104962")}.

See Also

EllCopSim() for the simulation of elliptical copula samples, EllCopEst() for the estimation of elliptical copula, conversion functions for the conversion between different representation of the generator of an elliptical copula.

ElliptCopulas documentation built on Sept. 11, 2024, 6:50 p.m.