Let $\{x_1,\ldots,x_N\}$ be observations drawn from an unknown (but certainly asymmetric) probability distribution.
I would like to find the probability distribution by using the KDE approach:
$$
\hat{f}(x) = \frac{1}{Nh}\sum_{i=1}^{N} K\bigl(\frac{x-x_i}{h}\bigr)
$$
However, I tried to use a Gaussian kernel, but it performed badly, since it is symmetric. Thus, I have seen that some work about the Gamma and Beta kernels have been released, although I did not understand how to operate with them.
My question is: how to handle this asymmetric case, supposing that the support of the underlying distribution is not in the interval $[0,1]$?
Best Answer
First of all, KDE with symmetric kernels can also work very well when you data is asymmetric. Otherwise, it would be completely useless in practice, actually.
Secondly, have you considered rescaling your data to fix the asymmetry, if you believe this is causing the problem. For example, it may be a good idea to try going to $\log(x)$, as this is known to help in many problems.