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

elldistrest

R Documentation

elldistrest

Description

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

Usage

elldistrest(
  Z,
  mu = 0,
  Sigma_m1,
  grid,
  h,
  Kernel = "epanechnikov",
  a,
  shrink,
  mpfr = FALSE,
  precBits = 100,
  dopb = FALSE,
  normalize = 1
)

Arguments

`Z`	A sample from a `q`-dimensional random vector `\mathbf{Z}` (`n \times q` matrix with observations in rows, variables in columns).
`mu`	The (estimated) mean of `\mathbf{Z}` (default `= 0`).
`Sigma_m1`	The (estimated) inverse of the scale matrix of `\mathbf{Z}`.
`grid`	The grid of values on which to estimate the density generator.
`h`	The bandwidth of the kernel.
`Kernel`	The kernel used for the smoothing (default = "epanechnikov").
`a`	The tuning parameter to improve the performance at `0`.
`shrink`	The shrinkage function to further improve the performance at `0` and guarantee the existence of the AMISE bandwidth.
`mpfr`	See the “EllDistrEst.R” function of the R package ‘ElliptCopulas’.
`precBits`	See the “EllDistrEst.R” function of the R package ‘ElliptCopulas’.
`dopb`	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{Z} following an elliptical distribution with generator g_{\mathcal{R}} : (0,\infty) \rightarrow \mathbb{R} and scale matrix \mathbf{R} such that the density of \mathbf{Z} is given by

h(\mathbf{z}) = \left |\mathbf{R} \right |^{-1/2} g_{\mathcal{R}} \left (\mathbf{z}^{\text{T}} \mathbf{R}^{-1} \mathbf{z} \right ),

for \mathbf{z} \in \mathbb{R}^{q}. Suppose that a sample \mathbf{Z}^{(1)}, \dots, \mathbf{Z}^{(n)} from \mathbf{Z} is given, and let \widehat{\mathbf{R}} be an estimator for the scale matrix \mathbf{R}. Then, when defining

\widehat{\mathbf{Y}}^{(\ell)} = \widehat{\mathbf{R}}^{-1/2} \mathbf{Z}^{(\ell)}

for \ell = 1, \dots, n, this function computes the estimator \widehat{g}_{\mathcal{R}}^{\text{I}} for g_{\mathcal{R}} given by

\widehat{g}_{\mathcal{R}}^{\text{I}}(t) = c^{\text{I}}(t) \sum_{\ell = 1}^{n} \left \{k \left (\frac{\psi(t) - \psi \left (\left | \left |\widehat{\mathbf{Y}}^{(\ell)} \right | \right |^{2} \right )}{h_{n} \alpha \left (\psi(t) \right )} \right ) + k \left (\frac{\psi(t) + \psi \left (\left | \left |\widehat{\mathbf{Y}}^{(\ell)} \right | \right |^{2} \right )}{h_{n} \alpha \left (\psi(t) \right )} \right ) \right \},

where c^{\text{I}}(t) = [\Gamma(q/2)/(\pi^{q/2} n h_{n} \alpha(\psi(t)))] t^{-q/2 + 1} \psi^{\prime}(t), with k the kernel and h_{n} the bandwidth. The function

\psi(t) = -a + \left (a^{q/2} + t^{q/2} \right )^{2/q},

with a > 0 a tuning parameter was introduced by Liebscher (2005), and the shrinkage function \alpha(t) yields further estimation improvement. We suggest to take (for q > 2)

\alpha(t) = 1 - \frac{1}{t^{\delta} + 1},

where \delta \in (3/4 - 1/q, 1) is another tuning parameter. When q = 2, one can just take \alpha(t) = 1, and the value of a does not matter.

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)})

# 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 = elldistrest(Z = sample, Sigma_m1 = diag(q), grid = grid, h = h, 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.