EllCopEst {ElliptCopulas}

Estimate the density generator of a (meta-)elliptical copula

Description

This function estimates the density generator of a (meta-)elliptical copula using the iterative procedure described in (Derumigny and Fermanian, 2022). This iterative procedure consists in alternating a step of estimating the data via Liebscher's procedure EllDistrEst() and estimating the quantile function of the underlying elliptical distribution to transform the data back to the unit cube.

Usage

EllCopEst(
  dataU,
  Sigma_m1,
  h,
  grid = seq(0, 10, by = 0.01),
  niter = 10,
  a = 1,
  Kernel = "epanechnikov",
  verbose = 1,
  startPoint = "identity",
  prenormalization = FALSE
)

Arguments

dataU

the data matrix on the [0,1] scale.

Sigma_m1

the inverse of the correlation matrix of the components of data

h

bandwidth of the kernel for Liebscher's procedure

grid

the grid at which the density generator is estimated.

niter

the number of iterations

a

tuning parameter to improve the performance at 0. See Liebscher (2005), Example p.210

Kernel

kernel used for the smoothing. Possible choices are gaussian, epanechnikov and triangular.

verbose

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

startPoint

is the given starting point of the procedure

• startPoint = "gaussian" for using the gaussian generator as starting point ;

• startPoint = "identity" for a data-driven starting point ;

• startPoint = "A~Phi^{-1}" for another data-driven starting point using the Gaussian quantile function.

prenormalization

if TRUE, the procedure will normalize the variables at each iteration so that the variance is 1.

Value

a list of two elements:

• g_d_norm: the estimated elliptical copula generator at each point of the grid;

• list_path_gdh: the list of estimated elliptical copula generator at each iteration.

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.

Liebscher, E. (2005). A semiparametric density estimator based on elliptical distributions. Journal of Multivariate Analysis, 92(1), 205. doi:10.1016/j.jmva.2003.09.007

See Also

EllDistrEst for the estimation of elliptical distributions, EllCopSim for the simulation of elliptical copula samples, EllCopLikelihood for the computation of the likelihood of a given generator, DensityGenerator.normalize to compute the normalized version of a given generator.

Examples

# Simulation from a Gaussian copula
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)
result = EllCopEst(dataU = U, grid, Sigma_m1 = diag(3),
                   h = 0.1, a = 0.5)
plot(grid, g_d, type = "l", xlim = c(0,2))
lines(grid, result$g_d_norm, col = "red", xlim = c(0,2))

# Adding missing observations
n_NA = 2
U_NA = U
for (i in 1:n_NA){
  U_NA[sample.int(n,1), sample.int(3,1)] = NA
}
resultNA = EllCopEst(dataU = U_NA, grid, Sigma_m1 = diag(3),
                     h = 0.1, a = 0.5)
lines(grid, resultNA$g_d_norm, col = "blue", xlim = c(0,2))