Epanechnikov distribution

Epanechnikov

Parameters	$c>0$ scale (real)
Support	$-c\leq x\leq c$
PDF	${\frac {3}{4c}}\max \left(0,1-\left({\frac {x}{c}}\right)^{2}\right)$
CDF	${\frac {1}{2}}+{\frac {3x}{4c}}-{\frac {x^{3}}{4c^{3}}}$ for $-c\leq x\leq c$
Mean	$0$
Median	$0$
Mode	$0$
Variance	${\frac {c^{2}}{5}}$

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:

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

where $c>0$ is a scale parameter. Setting $c={\sqrt {5}}$ gives the unit variance probability distribution originally considered by Epanechnikov.

Properties

Cumulative distribution function

The cumulative distribution function (CDF) of the Epanechnikov distribution is:

$F(x)={\frac {1}{2}}+{\frac {3x}{4c}}-{\frac {x^{3}}{4c^{3}}}$ for $-c\leq x\leq c$

Moments and other properties

• Mean: $E[X]=0$
• Median: ${\text{Median}}[X]=0$
• Mode: ${\text{Mode}}[X]=0$
• Variance: ${\text{Var}}[X]={\frac {c^{2}}{5}}$

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.