I was just wondering why there are 2 definitions of the Epanechnikov-Kernel. In the first paper, Epanechnikov introduced his kernel [1] with:

$Kleft(yright) = frac{3}{4 sqrt{5}} – frac{3 y^2}{20 sqrt{5}}$ for $|y| leq sqrt{5}$

In the lecture nodes of my course (+ wikipedia) i found following definition:

$Kleft(yright) = frac{3}{4} (1 – y^2)$ for $|y| leq 1$

Is the second one the general case, since we can adjust the size of $y$ with the bandwidth parameter?

Thanks in advance!

[1] http://www.mathnet.ru/links/57ece7df1c1ea5e1484f9abe34a7aefc/tvp1130.pdf :

Epanechnikov, V. A. "Nonparametric estimation of a multidimensional probability density." Teoriya veroyatnostei i ee primeneniya 14.1 (1969): 156-161.

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#### Best Answer

Let's distinguish the two kernels in our notation so we're not using the same symbol for each. So we have

$K^{(e)}left(yright) = frac{3}{4 sqrt{5}} – frac{3 y^2}{20 sqrt{5}}$ for $|y|leqsqrt{5}$

and

$K(y)=frac34 (1−y^2)$ for $|y|leq 1$

Consider $K_h(y) = frac{1}{h} K(frac{y}{h}),,text{ for }|frac{y}{h}|leq 1$.

By choosing $h=sqrt{5}$ we get

$K_sqrt{5}(y) = frac{1}{sqrt{5}} cdot frac34 (1−frac{y^2}{5})=frac{3}{4sqrt{5}} −frac{3y^2}{20sqrt{5}},, text{ for }|frac{y}{sqrt{5}}|leq 1$

so they're both the same but with different "base" bandwidth.

Is the second one the general case, since we can adjust the size of y with the bandwidth parameter?

Either may be adjusted with a bandwidth parameter, so they are exactly as general as each other.

As you suggest in comments under your question, this can certainly lead to confusion if you're reading a source that doesn't explicitly define its base (unit bandwidth) kernel, and also doesn't give a reference from which you can obtain it.

I have run into this problem a few times (though usually you can figure it out).

It underlines the importance, when you give a bandwidth, of being explicit about what the unit bandwidth case is.

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