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 kernels functions are commonly used: uniform, triangle, epanechnikov, quartic (biweight), tricube (triweight), gaussian, and cosine.

Below, the notation $1_{(p)}\,\!$ denotes the value 1 when p holds, and 0 when p is false.

Kernel Functions

Uniform	$K(u)={\frac {1}{2}}\ 1_{(|u|\leq 1)}$
Triangle	$K(u)=(1-|u|)\ 1_{(|u|\leq 1)}$
Epanechnikov	$K(u)={\frac {3}{4}}(1-u^{2})\ 1_{(|u|\leq 1)}$
Quartic	$K(u)={\frac {15}{16}}(1-u^{2})^{2}\ 1_{(|u|\leq 1)}$
Triweight	$K(u)={\frac {35}{32}}(1-u^{2})^{3}\ 1_{(|u|\leq 1)}$
Gaussian	$K(u)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}u^{2}}$
Cosine	$K(u)={\frac {\pi }{4}}\cos \left({\frac {\pi }{2}}u\right)1_{(|u|\leq 1)}$

All of the above Kernels in a Common Coordinate System

See also

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

References

Li, Qi (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press. ISBN 0691121613. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Kernel Basis function (with graphs).