Kernel (statistics)

A kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series, in the use of the periodogram to estimate the spectral density. An additional use is in the estimation of a time-varying intensity for a point process.

Commonly, kernel widths must also be specified when running a non-parametric estimation.

Definition

A kernel is a non-negative real-valued integrable function K satisfying the following two requirements:

• $\int_{-\infty}^{+\infty}K(u)\,du = 1\,;$
• $K(-u) = K(u) \mbox{ for all values of } u\,.$

The first requirement ensures that the method of kernel density estimation results in a probability density function. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.

If K is a kernel, then so is the function K* defined by K*(u) = λ⁻¹K(λ⁻¹u), where λ > 0. This can be used to select a scale that is appropriate for the data.

Kernel functions in common use

Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov, quartic (biweight), tricube, triweight, Gaussian, and cosine.

In the table below, 1_{…} is the indicator function.

Kernel Functions, K(u)			$\textstyle \int u^2K(u)du$	$\textstyle \int K^2(u)du$
Uniform	$K(u) = \frac12 \,\mathbf{1}_{\{|u|\leq1\}}$		$\frac13$	$\frac12$
Triangular	$K(u) = (1-|u|) \,\mathbf{1}_{\{|u|\leq1\}}$		$\frac{1}{6}$	$\frac{2}{3}$
Epanechnikov	$K(u) = \frac{3}{4}(1-u^2) \,\mathbf{1}_{\{|u|\leq1\}}$		$\frac{1}{5}$	$\frac{3}{5}$
Quartic (biweight)	$K(u) = \frac{15}{16}(1-u^2)^2 \,\mathbf{1}_{\{|u|\leq1\}}$		$\frac{1}{7}$	$\frac{5}{7}$
Triweight	$K(u) = \frac{35}{32}(1-u^2)^3 \,\mathbf{1}_{\{|u|\leq1\}}$		$\frac{1}{9}$	$\frac{350}{429}$
Tricube	$K(u) = \frac{70}{81}(1- {\left| u \right|}^3)^3 \,\mathbf{1}_{\{|u|\leq1\}}$		$\frac{35}{243}$	$\frac{175}{247}$
Gaussian	$K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^2}$		$1\,$	$\frac{1}{2\sqrt\pi}$
Cosine	$K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right) \mathbf{1}_{\{|u|\leq1\}}$		$1-\frac{8}{\pi^2}$	$\frac{\pi^2}{16}$

All of the above Kernels in a Common Coordinate System

See also

• Kernel density estimation
• Kernel smoother
• Stochastic kernel
• Density estimation

References

• Li, Qi; Racine, Jeffrey S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press. ISBN 0691121613.