ellcopest: ellcopest
In VecDep: Measuring Copula-Based Dependence Between Random Vectors

ellcopest

R Documentation

ellcopest

Description

This functions performs improved kernel density estimation of the generator of a meta-elliptical copula by using Liebscher's algorithm, combined with a shrinkage function.

Usage

ellcopest(
  dataU,
  Sigma_m1,
  h,
  grid,
  niter = 10,
  a,
  Kernel = "epanechnikov",
  shrink,
  verbose = 1,
  startPoint = "identity",
  prenormalization = FALSE,
  normalize = 1
)

Arguments

`dataU`	The (estimated) copula observations from a `q`-dimensional random vector `\mathbf{X}` (`n \times q` matrix with observations in rows, variables in columns).
`Sigma_m1`	The (estimated) inverse of the scale matrix of the meta-elliptical copula.
`h`	The bandwidth of the kernel.
`grid`	The grid of values on which to estimate the density generator.
`niter`	The number of iterations used in the MECIP (default = 10).
`a`	The tuning parameter to improve the performance at `0`.
`Kernel`	The kernel used for the smoothing (default = "epanechnikov").
`shrink`	The shrinkage function to further improve the performance at `0` and guarantee the existence of the AMISE bandwidth.
`verbose`	See the “EllDistrEst.R” function of the R package ‘ElliptCopulas’.
`startPoint`	See the “EllDistrEst.R” function of the R package ‘ElliptCopulas’.
`prenormalization`	See the “EllDistrEst.R” function of the R package ‘ElliptCopulas’.
`normalize`	A value in `\{1,2\}` indicating the normalization procedure that is applied to the estimated generator (default = 1).

Details

The context is the one of a q-dimensional random vector \mathbf{X} = (\mathbf{X}_{1}, \dots, \mathbf{X}_{k}),

with \mathbf{X}_{i} = (X_{i1}, \dots, X_{id_{i}}) for i = 1, \dots, k, having a meta-elliptical copula. This means that there exists a generator g_{\mathcal{R}} : (0,\infty) \rightarrow \mathbb{R} and a quantile function Q, such that the random vector \mathbf{Z} = (\mathbf{Z}_{1}, \dots, \mathbf{Z}_{k}) with

\mathbf{Z}_{i} = (Z_{i1}, \dots, Z_{id_{i}}) = \left(\left (Q \circ F_{i1} \right ) \left (X_{i1} \right ), \dots, \left (Q \circ F_{id_{i}} \right ) \left (X_{id_{i}} \right ) \right )

for i = 1, \dots, k, where F_{ij} is the cdf of X_{ij}, has a multivariate elliptical distribution. Denoting \widehat{F}_{ij}(x_{ij}) = \frac{1}{n+1} \sum_{\ell = 1}^{n} 1 \left (X_{ij}^{(\ell)} \leq x_{ij} \right ) for the (rescaled) empirical cdf of X_{ij} based on a sample X_{ij}^{(1)}, \dots, X_{ij}^{(n)} for i = 1, \dots, k and j = 1, \dots, d_{i}, and \widehat{\mathbf{R}} for an estimator of the scale matrix \mathbf{R}, this function estimates g_{\mathcal{R}} by using the MECIP (Meta-Elliptical Copula Iterative Procedure) of Derumigny & Fermanian (2022).

This means that we start from an initial guess \widehat{g}_{\mathcal{R},0} for the generator g_{\mathcal{R}}, based on which we obtain an estimated sample from \mathbf{Z} through the quantile function corresponding to \widehat{g}_{\mathcal{R},0}. Based on this estimated sample, we then obtain an estimator \widehat{g}_{\mathcal{R},1} using the function elldistrest, performing improved kernel estimation with shrinkage function. This procedure is repeated for a certain amount (niter) of iterations to obtain a final estimate for g_{\mathcal{R}}.

The estimator without the shrinkage function \alpha is implemented in the R package ‘ElliptCopulas’. We use this implementation and bring in the shrinkage function.

In order to make g_{\mathcal{R}} identifiable, an extra normalization procedure is implemented in line with an extra constraint on g_{\mathcal{R}}. When normalize = 1, this corresponds to \mathbf{R} being the correlation matrix of \mathbf{Z}. When normalize = 2, this corresponds to the identifiability condition of Derumigny & Fermanian (2022).

Value

The estimates for g_{\mathcal{R}} at the grid points.

References

Derumigny, A., Fermanian, J.-D., Ryan, V., van der Spek, R. (2024). ElliptCopulas, R package version 0.1.4.1. url: https://CRAN.R-project.org/package=ElliptCopulas.

Derumigny, A. & Fermanian, J.-D. (2022). Identifiability and estimation of meta-elliptical copula generators. Journal of Multivariate Analysis 190:104962.
doi: https://doi.org/10.1016/j.jmva.2022.104962.

De Keyser, S. & Gijbels, I. (2024). Hierarchical variable clustering via copula-based divergence measures between random vectors. International Journal of Approximate Reasoning 165:109090. doi: https://doi.org/10.1016/j.ijar.2023.109090.

Liebscher, E. (2005). A semiparametric density estimator based on elliptical distributions. Journal of Multivariate Analysis 92(1):205-225. doi: https://doi.org/10.1016/j.jmva.2003.09.007.

Examples


q = 4

# Sample size
n = 1000

# Grid on which to evaluate the elliptical generator
grid = seq(0.005,100,by = 0.005)

# Degrees of freedom
nu = 7

# Student-t generator with 7 degrees of freedom
g_q = ((nu/(nu-2))^(q/2))*(gamma((q+nu)/2)/(((pi*nu)^(q/2))*gamma(nu/2))) *
      ((1+(grid/(nu-2)))^(-(q+nu)/2))

# Density of squared radius
R2 = function(t,q){(gamma((q+nu)/2)/(((nu-2)^(q/2))*gamma(nu/2)*gamma(q/2))) *
                   (t^((q/2)-1)) * ((1+(t/(nu-2)))^(-(q+nu)/2))}

# Sample from 4-dimensional Student-t distribution with 7 degrees of freedom
# and identity covariance matrix
sample = ElliptCopulas::EllDistrSim(n,q,diag(q),density_R2 = function(t){R2(t,q)})

# Copula pseudo-observations
pseudos = matrix(0,n,q)
for(j in 1:q){pseudos[,j] = (n/(n+1)) * ecdf(sample[,j])(sample[,j])}

# Shrinkage function
shrinkage = function(t,p){1-(1/((t^p) + 1))}

# Tuning parameter selection
opt_parameters = elliptselect(n,q,seq((3/4)-(1/q)+0.01,1-0.01,len = 200),
                                  seq(0.01,2,len = 200))

# Optimal tuning parameters
a = opt_parameters$Opta ; p = opt_parameters$Optp ; h = opt_parameters$Opth

# Estimated elliptical generator
g_est = ellcopest(dataU = pseudos,Sigma_m1 = diag(q),h = h,grid = grid,a = a,
                  shrink = function(t){shrinkage(t,p)})

plot(grid,g_est,type = "l", xlim = c(0,8))
lines(grid,g_q,col = "green")

VecDep documentation built on April 4, 2025, 5:14 a.m.