DensityGenerator.normalize {ElliptCopulas} R Documentation

## 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. doi:10.1016/j.jmva.2022.104962.

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.