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Given an n-dimensional sample $\left\lbrace X_{i}\right\rbrace$ of i.i.d. observations, let's consider a kernel density function $$\hat{f}_{h_{n},n}(x)=\dfrac{1}{nh_{n}}\cdot\sum_{i=1}^n K\left(\dfrac{X_{i}-x}{h_{n}}\right)$$ where:

  • $h_{n}$ is a smoothing parmeter
  • $K(u)=\mathbb{1}\left( u\in\left\lbrace -1/2,1/2\right\rbrace \right)$ is an indicator function

Question:

what's the optimal smoothing parameter $h_{n}$ in this case?

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The answer will depend on your criterion for "optimal" and on the true density $f$ that is to be estimated. A common criterion is the Mean Integrated Square Error $$MISE = E\left( \int \left(f- \hat{f}\right)^2 dx\right)$$ where $\hat{f}$ is the kernel density estimate of $f$. One approach for finding an optimal $h$ is by means of "AMISE", which is an asymptotic epxansion (for small $h$) of MISE, and then analytically finding its minimum through $AMISE'(h) = 0$. This yields $$h_{AMISE} = \left[\frac{\int K^2 dx}{n \int \left(f''\right)^2 dx \int x^2 K^2 dx}\right]^2$$ This is the startng points for most choices of $h$, e.g., by inserting a Gaussian for $f$, or by inserting the estimate $\hat{f}$ for $f$ and solving the resulting implicit equation for $h$ ("plug-in method").

For a gentle introdutcion into this topic, see

M. C. Jones, J. S. Marron, S. J. Sheather: "A Brief Survey of Bandwidth Selection for Density Estimation. Journal of the American Statistical Association 91,433, pp. 401-407 (1996)

or for a not so gentle, but more recent overview

SJ. Sheather: "Density Estimation." Statistical Science 19,4, pp. 588–597 (2004)

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  • $\begingroup$ thank you @cdalitz ! Could you explain to me how to replace $f$ with $\hat{f}$, in this case, please? Since the uniform kernel density is not differentiable we cannot compute the second-order derivative of $\hat{f}$. Is there a way to obtain a pseudo derivative? thank you $\endgroup$ – Almostsurely Jul 3 '20 at 23:21
  • $\begingroup$ @Almostsurely: You are right that$\hat{f}$ is twice differentientiable only if $K$ is twice differentiable (the derivative can be pulled under the sum in the kernel density estimator). Obviously, this fails for your kernel representing a uniform distribution. In this case, you can use "Silverman's rule of thumb" instead, as described in the above articles. $\endgroup$ – cdalitz Jul 4 '20 at 8:14

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