This package implements several functions for the estimation of meta-elliptical copulas and for the estimation of elliptical and trans-elliptical distributions:

• elliptical distributions are distributions for which the isodensity surfaces/curves are ellipses. Their distribution is determined by a mean, a variance matrix and a univariate function called the generator.

• (meta-)elliptical copulas are copulas defined implicitly as copula functions of elliptical distributions. Their distribution is determined by a correlation matrix and a generator.

• trans-elliptical distributions are distributions whose copula is meta-elliptical and whose margins are arbitrary. In other words, a trans-elliptical distribution is a multivariate distribution built from the dependence structure (copula) of an elliptical distribution, but which could have any margin. Their distribution is therefore determined by the marginal distributions, the correlation matrix and the generator.

How to install

The release version on CRAN:

install.packages("ElliptCopulas")

The development version from GitHub:

# install.packages("remotes")
remotes::install_github("AlexisDerumigny/ElliptCopulas")

Main functions of the package

1. Inference of Elliptical Distributions

• EllDistrSim: simulate data from an elliptical distribution with a given arbitrary generator.

# Sample from an Elliptical distribution for which the
# squared radius follows an exponential distribution
mu = c(2,6,-5)
cov1 = rbind(c(1  , 0.3, 0.3),
             c(0.3, 1  , 0.3),
             c(0.3, 0.3, 1  ))
# cov1 = diag(3)
grid = seq(0,10, by = 0.1)
generator = exp(- grid/2) / (2*pi)^(3/2)
density_R2 = Convert_gd_To_fR2(grid = grid, g_d = generator, d = 3)
X = EllDistrSim(n = 1000, d = 3, A = chol(cov1), mu = mu,
                density_R2 = density_R2)
plot(X[, 1], X[, 2])

• EllDistrEst: nonparametric estimation of the generator of an elliptical distribution.

estDensityGenerator = EllDistrEst(X = X, mu = mu, Sigma_m1 = solve(cov1),
                                  grid = grid, a = 10, h = 0.02, dopb = FALSE)
plot(grid, estDensityGenerator, type = "l", ylab = "Estimated & true density generators")
lines(grid, generator, col = "red")

• EllDistrDerivEst: nonparametric estimation of the derivatives of the generator of an elliptical distribution.

• EllDistrEst.adapt: adaptive nonparametric estimation of the generator of an elliptical distribution.

2. Estimation of correlation matrix

• KTMatrixEst: fast estimation of Kendall’s tau correlation matrix assuming that it has a block structure. This procedure works even if the distribution is not elliptical.

However, in the elliptical case, it can be used to recover the (usual) Pearson’s correlation matrix for elliptical distribution, as both are then linked by the relationship