I have been doing work on fourth-order kernels and density estimation but was asked what sort of penalty I could put on the density estimate of the PDF going negative because of the nature of fourth-order kernels. Does anybody have any suggestions?
Yea sorry I was a bit vague. Using kernels to estimate density functions as in here Kernel density estimation - Wikipedia, the free encyclopedia . But using a higher order kernel in the case I am doing a fourth-order kernel you can get a higher rate of convergence by relaxing the condition that the kernel is itself a pdf and allowing it to take negative values.
This is now equal to 0 but x^4 K(x) is not equal to zero. So the fact that the kernel is negative in some places means that for small sample sizes the estimate of the pdf will be negative in some places. This is what I am looking to penalise maybe in terms of ISE (Integrated Squared Error) or something like that but I don't really have any ideas.