Kernel (statistics)

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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.


Image:Kerneldensityestimation.gif
Example of kernel density estimation (upper curve) as a sum of kernel functions (lower curves) each divided by Nh (Number of observations times the bandwidth)

Contents

[edit] Kernel functions

Several types of kernels functions are commonly used: uniform, triangle, epanechnikov, quartic(biweight), tricube (triweight), gaussian, cosinus.

[edit] Uniform

K(u) = \frac{1}{2}\ 1_{(|u|\leq1)}

[edit] Triangle

K(u) = (1-|u|)\ 1_{(|u|\leq1)}

[edit] Epanechnikov

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

[edit] Quartic

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

[edit] Triweight

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

[edit] Gaussian

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

[edit] Cosinus

K(u) = \frac{\pi}{4}\cos\left(\frac{\pi}{2}u\right)1_{(|u|\leq1)}

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[edit] Munaf

K(u)=\frac{45}{64}(1-u^2)^4\ 1_{(|u|\leq1)}

[edit] See also

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