EllDistrEst {ElliptCopulas}

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.

Usage

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

Arguments

X

matrix of observations.

mu

(estimated) mean of X.

Sigma_m1

(estimated) inverse of the covariance matrix of X.

grid

grid of values on which to estimate the density generator

h

bandwidth of the kernel

Kernel

kernel used for the smoothing

a

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

mpfr

if mpfr = TRUE, multiple precision floating point is set. This allows for a higher accuracy, at the expense of computing times. 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

if dopb = TRUE, a progressbar is displayed.

Value

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

Author(s)

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. doi:10.1016/j.jmva.2003.09.007

See Also

EllDistrSim for the simulation of elliptical distribution samples, EllCopEst for the estimation of elliptical copulas.

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

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")