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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	${\displaystyle {\frac {1}{2}}+{\frac {3x}{4c}}-{\frac {x^{3}}{4c^{3}}}}$ for ${\displaystyle -c\leq x\leq c}$
Mean	${\displaystyle 0}$
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle {\frac {c^{2}}{5}}}$

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 ^[1].

Definition

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

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

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

Properties

Cumulative distribution function

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

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

Moments and other properties

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

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.

References

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