Epanechnikov distribution explained

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:

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

where

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

Applications

The Epanechnikov distribution has applications in various fields, including:

• 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.

Notes and References

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