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I'm studying KDE and got trouble understanding Scott's rule or Silverman's rule for bandwidth selection.

I saw that the optimal bandwidth is the value that minimizes Mean Integrated Squared Error (MISE).

$MISE = \int E(\hat f(x)-f(x) )^2 d{x} $

But MISE formula can't be used directly since they involve the unknown, real density function $f(x)$.

Therefore, Scott's or Silverman's rule of thumb assumes Gaussian distribution for the unknown density $f(x)$ in order to find optimal bandwidth.

My doubt/question is:

  1. Non-parametric method like KDE is a distribution-free method, which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions. But since Scott or Silverman's rule assumes Gaussian distribution for the unknown density $f(x)$, it seems like a contradiction for me.

  2. Does Scott or Silverman's rule assumes that the kernel function to estimate density is also Gaussian?

Many thanks :)

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  • $\begingroup$ Scott’s rule assumes $h_n$ to be propor-tional to $d·n^{\frac{-1}{ m+4}}$ where $d$ is the standard deviation of the time series, $m$ is dimension of $x$ $\endgroup$
    – Nick
    Commented Nov 23, 2022 at 15:58

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