# "Efficiency" of a Kernel

My understanding is that the Epanechnikov kernel is "efficient" in a mean squared error sense. Footnote 4 of Wikipedia's page defines the "efficiency" of a kernel as $$\sqrt{\int u^2 K(u)du} \int K(u)^2 du.$$

which is minimized by the Epanechnikov kernel. (I've seen this definition in places other than Wikipedia). I am struggling to map this definition directly into mean squared error.

For kernel density estimation of the density of $$X$$, the mean squared error I derived is,

$$\frac{1}{4} h^4 \left(∫_{-∞}^∞z^2 K(z)dz\right)^2 f^{''}(x)^2+\frac{1}{nh} f(x) \left(∫_{-∞}^∞ K(z)^2 dz\right)$$

For a local constant (Nadaraya-Watson) regression estimating $$E[Y|X]$$, my derivation of MSE results in: $$h^4 \left(\int u^2 K(u)du\right)^2 B^2 (x)+\frac{\sigma^2(x)(\int K(u)^2 du))}{nhf(x)}$$

Where $$\sigma^2(x)$$ is the variance of $$Y$$ at $$X=x$$.

Mean Integrated Squared Error would involve integrating over $$x$$.

Can anyone help me understand how the definition of "efficiency" as posted on Wikipedia is directly related to minimizing MSE? (or MISE/AMISE?) The components of Wikipedia's definition definitely appear in MSE, but it definitely seems distinct.

TIA!

## 1 Answer

Kernels can be normalized based on $$\int K(u)^2 du$$. i.e. one could normalize such that $$\int K(u)^2 du =1$$, and the only term that affects efficiency is $$\int u^2 K(u) du$$.

Thus, mulitplying by $$\int K(u)^2 du$$ is just scaling based on the variance of the kernel (which could be normalized).