ElliptCopulas (version 0.1.4.1)

EllDistrEst: Nonparametric estimation of the density generator of an elliptical distribution

Description

This function uses Liebscher's algorithm to estimate the density generator of an elliptical distribution by kernel smoothing. A continuous elliptical distribution has a density of the form $$f_X(x) = {|\Sigma|}^{-1/2} g\left( (x-\mu)^\top \, \Sigma^{-1} \, (x-\mu) \right), $$ where \(x \in \mathbb{R}^d\), \(\mu \in \mathbb{R}^d\) is the mean, \(\Sigma\) is a \(d \times d\) positive-definite matrix and a function \(g: \mathbb{R}_+ \rightarrow \mathbb{R}_+\), called the density generator of \(X\). The goal is to estimate \(g\) at some point \(\xi\), by $$ \widehat{g}_{n,h,a}(\xi) := \dfrac{\xi^{\frac{-d+2}{2}} \psi_a'(\xi)}{n h s_d} \sum_{i=1}^n K\left( \dfrac{ \psi_a(\xi) - \psi_a(\xi_i) }{h} \right) + K\left( \dfrac{ \psi_a(\xi) + \psi_a(\xi_i) }{h} \right), $$ where \(s_d := \pi^{d/2} / \Gamma(d/2)\), \(\Gamma\) is the Gamma function, \(h\) and \(a\) are tuning parameters (respectively the bandwidth and a parameter controlling the bias at \(\xi = 0\)), \(\psi_a(\xi) := -a + (a^{d/2} + \xi^{d/2})^{2/d},\) \(\xi \in \mathbb{R}\), \(K\) is a kernel function and \(\xi_i := (X_i - \mu)^\top \, \Sigma^{-1} \, (X_i - \mu), \) for a sample \(X_1, \dots, X_n\).

Usage

EllDistrEst(
  X,
  mu = 0,
  Sigma_m1 = diag(d),
  grid,
  h,
  Kernel = "epanechnikov",
  a = 1,
  mpfr = FALSE,
  precBits = 100,
  dopb = TRUE
)

Value

the values of the density generator of the elliptical copula, estimated at each point of the grid.

Arguments

X

a matrix of size \(n \times d\), assumed to be \(n\) i.i.d. observations (rows) of a \(d\)-dimensional elliptical distribution.

mu

mean of X. This can be the true value or an estimate. It must be a vector of dimension \(d\).

Sigma_m1

inverse of the covariance matrix of X. This can be the true value or an estimate. It must be a matrix of dimension \(d \times d\).

grid

grid of values of \(\xi\) at which we want to estimate the density generator.

h

bandwidth of the kernel. Can be either a number or a vector of the size length(grid).

Kernel

name of the kernel. Possible choices are "gaussian", "epanechnikov", "triangular".

a

tuning parameter to improve the performance at 0. Can be either a number or a vector of the size length(grid). If this is a vector, the code will need to allocate a matrix of size nrow(X) * length(grid) which can be prohibitive in some cases.

mpfr

if mpfr = TRUE, multiple precision floating point is used via the package Rmpfr. This allows for a higher (numerical) accuracy, at the expense of computing time. It is recommended to use this option for higher dimensions.

precBits

number of precBits used for floating point precision (only used if mpfr = TRUE).

dopb

a Boolean value. If dopb = TRUE, a progress bar is displayed.

Author

Alexis Derumigny, Rutger van der Spek

References

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

The function \(\psi_a\) is introduced in Liebscher (2005), Example p.210.

See Also

• EllDistrSim for the simulation of elliptical distribution samples.

• estim_tilde_AMSE for the estimation of a component of the asymptotic mean-square error (AMSE) of this estimator \(\widehat{g}_{n,h,a}(\xi)\), assuming \(h\) has been optimally chosen.

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

• EllDistrDerivEst for the nonparametric estimation of the derivatives of the generator.

• EllCopEst for the estimation of elliptical copulas density generators.

Examples

# Comparison between the estimated and true generator of the Gaussian distribution
X = matrix(rnorm(500*3), ncol = 3)
grid = seq(0,5,by=0.1)
g_3 = EllDistrEst(X = X, grid = grid, a = 0.7, h=0.05)
g_3mpfr = EllDistrEst(X = X, grid = grid, a = 0.7, h=0.05,
                      mpfr = TRUE, precBits = 20)
plot(grid, g_3, type = "l")
lines(grid, exp(-grid/2)/(2*pi)^{3/2}, col = "red")

# In higher dimensions
# \donttest{
d = 250
X = matrix(rnorm(500*d), ncol = d)
grid = seq(0, 400, by = 25)
true_g = exp(-grid/2) / (2*pi)^{d/2}

g_d = EllDistrEst(X = X, grid = grid, a = 100, h=40)

g_dmpfr = EllDistrEst(X = X, grid = grid, a = 100, h=40,
                      mpfr = TRUE, precBits = 10000)
ylim = c(min(c(true_g, as.numeric(g_dmpfr[which(g_dmpfr>0)]))),
         max(c(true_g, as.numeric(g_dmpfr)), na.rm=TRUE) )
plot(grid, g_dmpfr, type = "l", col = "red", ylim = ylim, log = "y")
lines(grid, g_d, type = "l")
lines(grid, true_g, col = "blue")
# }