Epanechnikov kernel ($K_E(u)=\frac{3}{4}(1-u^2)1_{\{|u|\le 1\}}$)

The Epanechnikov kernel for density estimation is a nonnegative symmetric truncated parabola with integral one, ${\mu }_2(K_E)=1/5$, and $R(K_E)=3/5$. It has the smallest optimized asymptotic mean integrated squared error among nonnegative second-order density kernels, up to rescaling. Its canonical kernel has scale $15^{1/5}$ and support $[-15^{1/5},15^{1/5}]$.

Fix the second moment $v>0$. Put $B=\sqrt{5v}$ and $E(u)=3(B^2-u^2{)}_{+}/(4B^3)$. For every nonnegative unit-integral kernel for density estimation $K$ with the same second moment and finite $R(K)$, the mass and second-moment constraints give $\int (B^2-u^2)(K-E)=0$. Outside $[-B,B]$, $E=0$ and $(B^2-u^2)K\le 0$, so $\int E(K-E)\ge 0$. Expanding the square gives $R(K)-R(E)=\int (K-E{)}^2+2\int E(K-E)\ge 0$, with equality only when $K=E$ almost everywhere. Since $R(K{)}^2{\mu }_2(K)$ is scale invariant, rescaled Epanechnikov kernels minimize the optimized asymptotic mean integrated squared error over all such second moments. This conclusion concerns nonnegative second-order kernels; signed higher-order kernels obey a different bias expansion.