Epanechnikov distribution

Epanechnikov

Parameters	${\displaystyle c>0}$ scale (real)
Support	${\displaystyle -c\leq x\leq c}$
PDF	${\displaystyle {\frac {3}{4c}}\max \left(0,1-\left({\frac {x}{c}}\right)^{2}\right)}$
CDF	${\displaystyle {\begin{cases}0&{\text{for }}x\leq -c\\{\frac {1}{2}}+{\frac {3x}{4c}}-{\frac {x^{3}}{4c^{3}}}&{\text{for }}x\in (-c,c)\\1&{\text{for }}x\geq c\end{cases}}}$
Mean	${\displaystyle 0}$
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle {\frac {c^{2}}{5}}}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle -{\frac {6}{7}}}$

In probability theory and statistics, the Epanechnikov distribution, also known as the Epanechnikov kernel, is a continuous probability distribution that is defined on a finite interval. It is named after V. A. Epanechnikov, who introduced it in 1969 in the context of kernel density estimation. ^[1]

Definition

A random variable has an Epanechnikov distribution if its probability density function is given by:

${\displaystyle p(x|c)={\frac {3}{4c}}\max \left(0,1-\left({\frac {x}{c}}\right)^{2}\right)}$

where ${\displaystyle c>0}$ is a scale parameter. Setting ${\displaystyle c={\sqrt {5}}}$ yields a unit variance probability distribution.

Applications

The Epanechnikov distribution has applications in various fields, including: Statistics, where it underpins optimal kernel density estimation for data smoothing; machine learning, enhancing techniques like anomaly detection; econometrics, aiding in the analysis of economic data distributions; signal processing, facilitating precise signal feature extraction; and image processing, contributing to advanced image smoothing and enhancement methods.

• Kernel density estimation: It is widely used as a kernel function in non-parametric statistics, particularly in kernel density estimation. In this context, it is often referred to as the Epanechnikov kernel. For more information, see Kernel functions in common use.

Related distributions

• The Epanechnikov distribution can be viewed as a special case of a Beta distribution that has been shifted and scaled along the x-axis.

References

1. Epanechnikov, V. A. (January 1969). "Non-Parametric Estimation of a Multivariate Probability Density". Theory of Probability & Its Applications. 14 (1): 153–158. doi:10.1137/1114019.