ElliptCopulas (version 0.1.4.1)

compute_etahat: Compute \(\hat{\eta}_k\)

Description

\(\hat{\eta}_k\) is a quantity that is useful for estimating the \(k\)-th derivative of the generator of an elliptical distribution. It is defined in Section 3 of (Ryan and Derumigny, 2024).

Usage

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

Value

a vector of size n1 = length(grid). Each component of this vector is \(\hat{\eta}_k(x[i])\)

where x[i] is the \(i\)-th element 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 on which 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.

k

order of the derivative

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

Victor Ryan, Alexis Derumigny

Examples

if (FALSE){
# Comparison between the estimated and true generator of the Gaussian distribution
n = 500000
d = 3
X = matrix(rnorm(n * d), ncol = d)
grid = seq(0, 5, by = 0.1)
a = 0.7

etahat = compute_etahat(X = X, grid = grid, a = a, h = 0.04, k = 1)
plot(grid, etahat, type = "l", ylim = c(-0.02, 0.02))

# Computation of true values
g = exp(-grid/2)/(2*pi)^{3/2}
gprime = (-1/2) *exp(-grid/2)/(2*pi)^{3/2}
A = a^(d/2)
psia = -a + (A + grid^(d/2))^(2/d)
psiaprime = grid^(d/2 - 1) * (A + grid^(d/2))^(2/d - 1)
psiasecond = psiaprime * ( (d-2)/2 ) * grid^{-1} * A *
  ( grid^(d/2) + A )^(-1)

rhoprimexi = ((d-2) * grid^((d-4)/2) * psiaprime
- 2 * grid^((d-2)/2) * psiasecond) / (2 * psiaprime^3) * g +
grid^((d-2)/2) / (psiaprime^2) * gprime

lines(grid, rhoprimexi, col = "red")
}