Epanechnikov core

Drawing of the Epanechnikov core

The Epanechnikov core (according to WA Jepanetschnikow) is the core that fulfills the following properties for a compact carrier :

1. ${\displaystyle k(x)\geq 0}$ for all ${\displaystyle x\in \mathbb {R} }$
2. ${\displaystyle \int k(x)\,{\mathrm {d} }x=1}$
3. ${\displaystyle \int x^{2}k(x)\,{\mathrm {d} }x=1}$
4. ${\displaystyle \int k^{2}(x)\,{\mathrm {d} }x}$ is minimized.

With these properties, the Epanechnikov kernel minimizes the mean square deviation of the corresponding kernel density estimator among all kernels . It is a polynomial of the form . ${\displaystyle a+bx^{2}}$

We want to put the numerical factors of the core in context. First consider the normalized family , whose terms assume a hill shape in the interval and which converges to the rectangular distribution of the height for large n : ${\displaystyle a,b}$ ${\displaystyle k_{n,d}(x)}$ ${\displaystyle [-d,d]}$ ${\displaystyle {\tfrac {1}{2d}}}$

${\displaystyle k_{n,d}(x)={\begin{cases}{\frac {1}{2d}}\left(1+{\frac {1}{2n}}\right)\left(1-\left({\frac {x}{d}}\right)^{2n}\right)&,|x|\leq d\\0&,|x|>d\end{cases}}}$

For this applies

${\displaystyle \int _{-\infty }^{\infty }x^{2n}k_{n,d}(x)\,{\mathrm {d} }x={\frac {d^{2n}}{4n+1}}.}$

The kernel given by Epanechnikov himself normalizes this integral for to one. So for we choose : ${\displaystyle n=1}$ ${\displaystyle \left({\tfrac {d}{\sqrt {4+1}}}\right)^{2}=1}$ ${\displaystyle k_{E}:=k_{1,{\sqrt {5}}}}$

${\displaystyle k_{E}(x)={\begin{cases}{\frac {3}{4{\sqrt {5}}}}\left(1-{\frac {x^{2}}{5}}\right)&,|x|\leq {\sqrt {5}}\\0&,|x|>{\sqrt {5}}\end{cases}}}$

Sometimes the core is also referred to as the Epanechnikov core, which accordingly does not meet property 3: ${\displaystyle d=1}$

${\displaystyle k_{E}(x)={\begin{cases}{\frac {3}{4}}(1-x^{2})&,|x|\leq 1\\0&,|x|>1\end{cases}}}$

Web links

• Proof of the properties of the Epanechnikov core (PDF file; 66 kB)

swell

1. ↑ VA Epanechnikov: Non-Parametric Estimation of a Multivariate Probability Density . In: Theory of Probability and its Applications , 1969, p. 156