TEllDistrEst {ElliptCopulas}

Estimation of trans-elliptical distributions

Description

This function estimates the parameters of a trans-elliptical distribution which is a distribution whose copula is (meta-)elliptical, with arbitrary margins, using the procedure proposed in (Derumigny & Fermanian, 2022).

Usage

TEllDistrEst(
  X, estimatorCDF = function(x){
    force(x)
    return( function(y){(stats::ecdf(x)(y) - 1/(2*length(x))) }) },
  h, verbose = 1, grid, ...)

Arguments

X

the matrix of observations of the variables

estimatorCDF

the way of estimating the marginal cumulative distribution functions. It should be either a function that takes in parameter a vector of observations and returns an estimated cdf (i.e. a function) or a list of such functions to be applied on the data. In this case, it is required that the length of the list should be the same as the number of columns of X. It is required that the functions returned by estimatorCDF should have values in the open interval (0,1).

h

bandwidth for the non-parametric estimation of the density generator.

verbose

if 1, prints the progress of the iterations. If 2, prints the normalizations constants used at each iteration, as computed by DensityGenerator.normalize.

grid

grid of values on which to estimate the density generator

...

other parameters to be passed to EllCopEst.

Value

This function returns a list with three components:

• listEstCDF: a list of estimated marginal CDF given by estimatorCDF;

• corMatrix: the estimated correlation matrix:

• estEllCopGen: the estimated generator of the meta-elliptical copula.

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.

Examples

cor = matrix(c(1, 0.5, 0.2,
               0.5, 1, 0.8,
               0.2, 0.8, 1), byrow = TRUE, nrow = 3)

grid = seq(0,10,by = 0.01)
g_d = DensityGenerator.normalize(grid, grid_g = exp(-grid), d = 3)
n = 10
# To have a nice estimation, we suggest to use rather n=200
# (around 20s of computation time)
U = EllCopSim(n = n, d = 3, grid = grid, g_d = g_d, A = chol(cor))
X = matrix(nrow = n, ncol = 3)
X[,1] = stats::qnorm(U[,1], mean = 2)
X[,2] = stats::qt(U[,2], df = 5)
X[,3] = stats::qt(U[,3], df = 8)

result = TEllDistrEst(X, h = 0.1, grid = grid)
plot(grid, g_d, type = "l", xlim = c(0,2))
lines(grid, result$estiEllCop$g_d_norm, col = "red")
print(result$corMatrix)

# Adding missing observations
n_NA = 2
X_NA = X
for (i in 1:n_NA){
  X_NA[sample.int(n,1), sample.int(3,1)] = NA
}
resultNA = TEllDistrEst(X_NA, h = 0.1, grid = grid, verbose = 1)
lines(grid, resultNA$estiEllCopGen, col = "blue")