The term kernel has several distinct meanings in statistics.
In Bayesian statistics
In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf. These factors form part of the normalization factor of the probability distribution, and are unnecessary in many situations. For example, in pseudorandom number sampling, most sampling algorithms ignore the normalization factor. In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, and if it matches a known distribution, the normalization factor can be reinstated. Otherwise, it may be unnecessary (for example, if the distribution only needs to be sampled from).
For many distributions, the kernel can be written in closed form, but not the normalization constant.
An example is the normal distribution. Its probability density function is

p(x\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{\frac{(x\mu)^2}{2\sigma^2}}
and the associated kernel is

p(x\mu,\sigma^2) \propto e^{\frac{(x\mu)^2}{2\sigma^2}}
Note that the factor in front of the exponential has been omitted, even though it contains the parameter \sigma^2 , because it is not a function of the domain variable x .
In pattern analysis
The kernel of a reproducing kernel Hilbert space is used in the suite of techniques known as kernel methods to perform tasks such as statistical classification, regression analysis, and cluster analysis on data in an implicit space. This usage is particularly common in machine learning.
In nonparametric statistics
In nonparametric statistics, a kernel is a weighting function used in nonparametric 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 timeseries, in the use of the periodogram to estimate the spectral density. An additional use is in the estimation of a timevarying intensity for a point process.
Commonly, kernel widths must also be specified when running a nonparametric estimation.
Definition
A kernel is a nonnegative realvalued 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) = λK(λu), 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,^{[1]} quartic (biweight), tricube,^{[2]} triweight, Gaussian,quadratic^{[3]} and cosine.
In the table below, 1_{{…}} is the indicator function.
Kernel Functions, K(u)

\textstyle \int u^2K(u)du

\textstyle \int K(u)^2 du

Efficiency* relative to the Epanechnikov kernel

Uniform

K(u) = \frac12 \,\mathbf{1}_{\{u\leq1\}}


\frac13

\frac12

1.076

Triangular

K(u) = (1u) \,\mathbf{1}_{\{u\leq1\}}


\frac{1}{6}

\frac{2}{3}

1.014

Epanechnikov

K(u) = \frac{3}{4}(1u^2) \,\mathbf{1}_{\{u\leq1\}}


\frac{1}{5}

\frac{3}{5}

1.000

Quartic
(biweight)

K(u) = \frac{15}{16}(1u^2)^2 \,\mathbf{1}_{\{u\leq1\}}


\frac{1}{7}

\frac{5}{7}

1.006

Triweight

K(u) = \frac{35}{32}(1u^2)^3 \,\mathbf{1}_{\{u\leq1\}}


\frac{1}{9}

\frac{350}{429}

1.013

Tricube

K(u) = \frac{70}{81}(1 {\left u \right}^3)^3 \,\mathbf{1}_{\{u\leq1\}}


\frac{35}{243}

\frac{175}{247}

1.002

Gaussian

K(u) = \frac{1}{\sqrt{2\pi}}e^{\frac{1}{2}u^2}


1\,

\frac{1}{2\sqrt\pi}

1.051

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}

1.0005

Logistic

K(u) = \frac{1}{e^{u}+2+e^{u}}


\frac{\pi^2}{3}

\frac{1}{6}

1.127

Silverman kernel^{[4]}

K(u) = \frac{1}{2} e^{\frac{u}{\sqrt{2}}}\cdot \sin\left( \frac{u}{\sqrt{2}}+\frac{\pi}{4}\right)


0

\frac{3\sqrt{2}}{16}

not applicable


Efficiency is defined as \left(\int u^2 K(u)d u\right)^{\frac{1}{2}}\cdot \int K(u)^2 d u.
Some of the kernels mentioned above in a common coordinate system
See also
References

^ Named for Epanechnikov, V. A. (1969). "NonParametric Estimation of a Multivariate Probability Density". Theory Probab. Appl. 14 (1): 153–158.

^ Altman, N. S. (1992). "An introduction to kernel and nearest neighbor nonparametric regression". The American Statistician 46 (3): 175–185.

^ Cleveland, W. S. & Devlin, S. J. (1988). "Locally weighted regression: An approach to regression analysis by local fitting". Journal of the American Statistical Association 83: 596–610.

^ Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.

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

Zucchini, Walter. "APPLIED SMOOTHING TECHNIQUES Part 1: Kernel Density Estimation". Retrieved 28 March 2012.
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