kernels {kdensity}

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

Be careful combining kernels with compact support with parametric starts, as the normalizing integral typically fails to converge. Use gaussian instead.

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 kernel of Chen (2000). For use on the positive half-line. gamma is the recommended biased-corrected kernel. 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 states the standard error of the kernel. This is used to make kernels comparable to the Gaussian kernel when calculating bandwidths.

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

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

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)
    exp(-inside/(2*(1-rho^2)))
  },
  support = c(0, 1)
)