kernelFun: Basic univatiate kernel functions

In smoothemplik: Smoothed Empirical Likelihood

Basic univatiate kernel functions

Description

Computes 5 most popular kernel functions of orders 2 and 4 with the potential of returning an analytical convolution kernel for density cross-validation. These kernels appear in \insertCitesilverman1986densitysmoothemplik.

Usage

kernelFun(
  x,
  kernel = c("gaussian", "uniform", "triangular", "epanechnikov", "quartic"),
  order = c(2, 4),
  convolution = FALSE
)

Arguments

x	A numeric vector of values at which to compute the kernel function.
kernel	Kernel type: uniform, Epanechnikov, triangular, quartic, or Gaussian.
order	Kernel order. 2nd-order kernels are always non-negative. *k*-th-order kernels have all moments from 1 to (k-1) equal to zero, which is achieved by having some negative values. \int_{-\infty}^{+\infty} x^2 k(x) = \sigma^2_k = 1. This is useful because in this case, the constant k_2 in formulæ 3.12 and 3.21 from \insertCitesilverman1986density;textualsmoothemplik is equal to 1.
convolution	Logical: return the convolution kernel? (Useful for density cross-validation.)

Details

The kernel functions take non-zero values on [-1, 1], except for the Gaussian one, which is supposed to have full support, but due to the rapid decay, is indistinguishable from machine epsilon outside [-8.2924, 8.2924].

Value

A numeric vector of the same length as input.

References

\insertAllCited

Examples

ks <- c("uniform", "triangular", "epanechnikov", "quartic", "gaussian"); names(ks) <- ks
os <- c(2, 4); names(os) <- paste0("o", os)
cols <- c("#000000CC", "#0000CCCC", "#CC0000CC", "#00AA00CC", "#BB8800CC")
put.legend <- function() legend("topright", legend = ks, lty = 1, col = cols, bty = "n")
xout <- seq(-4, 4, length.out = 301)
plot(NULL, NULL, xlim = range(xout), ylim = c(0, 1.1),
  xlab = "", ylab = "", main = "Unscaled kernels", bty = "n"); put.legend()
for (i in 1:5) lines(xout, kernelFun(xout, kernel = ks[i]), col = cols[i])
oldpar <- par(mfrow = c(1, 2))
plot(NULL, NULL, xlim = range(xout), ylim = c(-0.1, 0.8), xlab = "", ylab = "",
  main = "4th-order kernels", bty = "n"); put.legend()
for (i in 1:5) lines(xout, kernelFun(xout, kernel = ks[i], order = 4), col = cols[i])
par(mfrow = c(1, 1))
plot(NULL, NULL, xlim = range(xout), ylim = c(-0.25, 1.4), xlab = "", ylab = "",
  main = "Convolution kernels", bty = "n"); put.legend()
for (i in 1:5) {
  for (j in 1:2) lines(xout, kernelFun(xout, kernel = ks[i], order = os[j],
  convolution = TRUE), col = cols[i], lty = j)
}; legend("topleft", c("2nd order", "4th order"), lty = 1:2, bty = "n")
par(oldpar)

# All kernels integrate to correct values; we compute the moments
mom <- Vectorize(function(k, o, m, c) integrate(function(x) x^m * kernelFun(x, k, o,
  convolution = c), lower = -Inf, upper = Inf)$value)
for (m in 0:4) {
  cat("\nComputing integrals of x^", m, " * f(x). \nSimple unscaled kernel:\n", sep = "")
  print(round(outer(os, ks, function(o, k) mom(k, o, m = m, c = FALSE)), 4))
  cat("Convolution kernel:\n")
  print(round(outer(os, ks, function(o, k) mom(k, o, m = m, c = TRUE)), 4))
}

smoothemplik documentation built on Aug. 22, 2025, 1:11 a.m.