Epanechnikov distribution

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.

Definition

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

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

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.