kernels: Kernel functions

In kdensity: Kernel Density Estimation with Parametric Starts and Asymmetric Kernels

Kernel functions

Description

Kernel functions are an important part of kdensity. This document lists the available built-in functions and the structure of them. Any kernel in the list can be used in kdensity by using kernel = "kernel" for the intended kernel.

Details

Ordinary symmetric kernel estimates use exact normalization constants. With a non-uniform parametric start, compact-support kernels still require numerical normalization, which can fail to converge; use gaussian if this occurs.

Symmetric kernels

⁠gaussian, normal⁠: The Gaussian kernel. The default argument when starts is supported on R. ⁠epanechnikov, rectangular (uniform), triangular, biweight, cosine, optcosine⁠: Standard symmetric kernels, also used in stats::density(). ⁠tricube, triweight⁠: Standard symmetric kernels. Not supported by stats::density(). laplace: Uses the Laplace density, also known as the double exponential density.

Asymmetric kernels

⁠gamma, gamma_biased⁠: The gamma kernels of Chen (2000). For use on the positive half-line. gamma_biased is Chen's first kernel, a gamma density with shape y / h + 1 and scale h. gamma is the recommended bias-corrected (modified) kernel, with shape y / h when ⁠y >= 2h⁠ and ⁠(y / (2h))^2 + 1⁠ otherwise. gcopula: The Gaussian copula kernel of Jones & Henderson (2007). For use on the unit interval. ⁠beta, beta_biased⁠: The beta kernel of Chen (1999). For use on the unit interval. beta is the recommended bias-corrected kernel.

Structure

A kernel is a list containing two mandatory elements and one optional element. The mandatory element 'kernel' is the kernel function. It takes arguments ⁠y, x, h⁠, where x is the data supplied to kdensity and y is the point of evaluation. h is the bandwidth. Internally, the kernel function is evaluated as 1/h*kernel(y, x, h). It should be vectorized in x, but vectorization in y is not needed.

The second mandatory element is support, stating the domain of definition for the kernel. This is used to distinguish kernels on the unit interval / positive half-line from kernels on R.

sd is used for symmetric kernels and is the reciprocal of the kernel's standard deviation. This scale factor converts a bandwidth on the Gaussian standard-deviation scale to h = adjust * bw * sd.

Custom objects must use names different from the built-ins (for example, my_gaussian). Registering another custom object under the same name replaces the earlier custom object.

References

Chen, Song Xi. "Probability density function estimation using gamma kernels." Annals of the Institute of Statistical Mathematics 52.3 (2000): 471-480. Jones, M. C., and D. A. Henderson. "Kernel-type density estimation on the unit interval." Biometrika 94.4 (2007): 977-984. Chen, Song Xi. "Beta kernel estimators for density functions." Computational Statistics & Data Analysis 31.2 (1999): 131-145.

See Also

kdensity(); parametric_starts(); bandwidths().

Examples

my_gaussian <- list(
  kernel = function(y, x, h) stats::dnorm((y - x) / h),
  sd = 1,
  support = c(-Inf, Inf)
)

my_gcopula <- list(
  kernel = function(y, x, h) {
    rho <- 1 - h^2
    inside <- rho^2 * (qnorm(y)^2 + qnorm(x)^2) - 2 * rho * qnorm(y) * qnorm(x)
    h / sqrt(1 - rho^2) * exp(-inside / (2 * (1 - rho^2)))
  },
  support = c(0, 1)
)

kdensity documentation built on Oct. 28, 2026, 5:06 p.m.