# Bandwidth selection for kernel density estimation, using a Weibull kernel

Let $\{s_1,\ldots,s_N\}$ be a collection of N samples. I have performed the kernel density estimation using the classical form: $$\hat{f}(x) = \frac{1}{Nh}\sum_{i=1}^N K\left(\frac{x-s_i}{h}\right)$$ where $K\left(\frac{x-s_i}{h}\right)$ is computed as a Weibull kernel, with $k=1$ and $\lambda=1$.

My question is: which is the easiest way to compute the bandwidth $h$? I am using the sample standard deviation, but it seems to be too small, i.e., the computed PDF appears too noisy at the end.

Thanks.

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You could probably try to use the "rule-of-thumb" for density estimation based on the sample size and the standard deviation: $$1.06\sigma n^{-\frac{1}{5}}$$