I've just started using Kernel Density Estimation for my study, and encountered a problem.
In KDE, we have to select a proper bandwidth $h$ according to the data. If we don't, it could lead wrong estimation as shown in the figure (taken from Wikipedia), where the gray line is a true PDF.
My intention is to get two large peaks (to be precisely, the one whose sample's value is smaller) from the density function because the data ideally/theoretically has two peaks. However, using
kde results in getting more than 2 peaks. It is said that the optimal bandwidth is
where $\hat\sigma$ is the standard deviation of the samples provided that the kernel function is Gaussian basis and that samples follow normal distribution. Since the density function should have two peak as mentioned above, the samples don't seem to follow normal distribution and thus don't meet the assumption.
So the question is: is it OK to choose a bigger bandwidth to smooth the density function and to decrease the number of peaks, or should I use the bandwidth which
kde selects by default and acquire the desired two peaks with some post-process?
It may be said that I could use either way as long as it returns the intended result, but I'm glad if there is some statistical evidence which supports the method.
I am very sorry that I cannot show the examples I'm actually working on due to sensitive concerns.
I appreciate any kinds of help. Thanks in advance.