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here https://stat.ethz.ch/R-manual/R-devel/library/stats/html/ksmooth.html is bandwidth explained as "the bandwidth. The kernels are scaled so that their quartiles (viewed as probability densities) are at +/- 0.25*bandwidth.". I don't understand it properly. Could someone please explain me more undrestandable what does it mean?

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The bandwidth is the "width" of the kernel function; larger bandwidths will give you a smoother estimate.

Often, if you're using a Gaussian kernel $K(x) = \exp\left( - \frac{x^2}{2 \sigma^2} \right)$, people refer to $\sigma$ (the "standard deviation") as the bandwidth.

In this case, they're using a slightly different definition in terms of the quartiles – the points at which the CDF of the distribution for which the kernel is the density are 0.25 or 0.75, i.e. the two values of $t$ where $\int_{-\infty}^{t} K(x) \,dx \in \{0.25, 0.75\}$. For a Gaussian distribution, the quartiles are at about $\frac{2}{3} \sigma$ away from the mean, so it seems this package is using "bandwidth" to mean four times that, $\frac{8}{3} \sigma \approx 2.67 \sigma$.

I don't know why they made that choice.

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