<p>Kernels can be normalized based on <span class="math-container">$\int K(u)^2 du$</span>. i.e. one could normalize such that <span class="math-container">$\int K(u)^2 du =1$</span>, and the only term that affects efficiency is the so-called &quot;roughness&quot; of the kernel, <span class="math-container">$\int u^2 K(u)^2 du $</span>.</p>
<p>Thus, mulitplying by <span class="math-container">$\int K(u)^2 du$</span> is just scaling based on the variance of the kernel (which could be normalized).</p>

CC BY-SA 4.0
2021-08-30T07:33:58.230Z
2021-08-30T07:33:58.230Z
Michael Gmeiner
<p>My understanding is that the Epanechnikov kernel is &quot;efficient&quot; in a mean squared error sense. Footnote 4 of <a href="https://en.wikipedia.org/wiki/Kernel_(statistics)" rel="nofollow noreferrer">Wikipedia's page</a> defines the &quot;efficiency&quot; of a kernel as <span class="math-container">$$\sqrt{\int u^2 K(u)du}  \int K(u)^2 du.$$</span></p>
<p>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.</p>
<p>For kernel density estimation of the density of <span class="math-container">$X$</span>, the mean squared error I derived is,</p>
<p><span class="math-container">$$\frac{1}{4} h^4 \left(∫_{-∞}^∞z^2 K(z)dz\right)^2 f^{''}(x)^2+\frac{1}{nh} f(x) \left(∫_{-∞}^∞ K(z)^2 dz\right)$$</span></p>
<p>For a local constant (Nadaraya-Watson) regression estimating <span class="math-container">$E[Y|X]$</span>, my derivation of MSE results in:  <span class="math-container">$$h^4 \left(\int u^2 K(u)du\right)^2 B^2 (x)+\frac{\sigma^2(x)(\int K(u)^2 du))}{nhf(x)} $$</span></p>
<p>Where <span class="math-container">$\sigma^2(x)$</span> is the variance of <span class="math-container">$Y$</span> at <span class="math-container">$X=x$</span>.</p>
<p>Mean Integrated Squared Error would involve integrating over <span class="math-container">$x$</span>.</p>
<p>Can anyone help me understand how the definition of &quot;efficiency&quot; 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.</p>
<p>TIA!</p>

<econometrics>
"Efficiency" of a kernel
Michael Gmeiner
# "Efficiency" of a kernel

(Asked by: username_0 on 2021-08-23)

My understanding is that the Epanechnikov kernel is "efficient" in a mean squared error sense. Footnote 4 of [Wikipedia's page](https://en.wikipedia.org/wiki/Kernel_(statistics)) 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!

**Tags:** econometrics

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## 1 answers

### Answer by username_0 on 2021-08-30. Score: 2 | Accepted 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 the so-called "roughness" of the kernel, $\int u^2 K(u)^2 du $.

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


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