ElliptCopulas (version 0.1.4.1)

EllDistrDerivEst: Estimate the derivatives of a generator

Description

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 the derivatives of \(g\) at some point \(\xi\), by kernel smoothing, following Section 3 of (Ryan and Derumigny, 2024).

Usage

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

Value

a matrix of size length(grid) * (kmax + 1)

with the estimated value of the generator and all its derivatives at all orders until and including kmax, at all points 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

highest order of the derivative of the generator that is to be estimated. For example, k = 1 corresponds to the estimation of the generator and of its derivative. k = 2 corresponds to the estimation of the generator as well as its first and second derivatives.

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, Victor Ryan

Victor Ryan, Alexis Derumigny

Details

Note that this function may be rather slow for higher-order derivatives. Furthermore, it is likely that the number of observations needs to be quite high for the higher-order derivatives to be estimated well enough.

References

Ryan, V., & Derumigny, A. (2024). On the choice of the two tuning parameters for nonparametric estimation of an elliptical distribution generator arxiv:2408.17087.

See Also

EllDistrEst for the nonparametric estimation of the elliptical distribution density generator itself, EllDistrSim for the simulation of elliptical distribution samples.

This function uses the internal functions compute_etahat and compute_matrix_alpha.

Examples

# Comparison between the estimated and true generator of the Gaussian distribution
n = 50000
d = 3
X = matrix(rnorm(n * d), ncol = d)
grid = seq(0, 5, by = 0.1)
a = 1.5

gprimeEst = EllDistrDerivEst(X = X, grid = grid, a = a, h = 0.09, k = 1)[,2]
plot(grid, gprimeEst, type = "l")

# Computation of true values
g = exp(-grid/2)/(2*pi)^{3/2}
gprime = (-1/2) * exp(-grid/2)/(2*pi)^{3/2}

lines(grid, gprime, col = "red")