Kernel Density Estimation for non-parametric

I'm writing an R function to get the fitted values of the kernel density estimate. For that I use the computational formula of summation of ({n-1 h-1 K{(x - Xi)/h}}?) $$\hat{f}(x) = \frac1{n h}\sum_{i=1}^n K\left(\frac{x-X_i}{h}\right)$$ where $$n$$ is the number of observations and $$h$$ is the bandwidth. Here $$K$$ is supposed to be the kernel function, but I don't find a clear formula to plug-in for $$K$$ in this formula.

I'm known that there are various kernel functions, but I clearly couldn't find the list of kernel functions that exist. (Epanechnikov kernel, cosine, Gaussian, Parzen, rectangular, and triangle kernels are among).

Could someone kindly provide me with a/some straightforward formulas to obtain K?

I used the following article:

• Well, you really need to learn $\LaTeX$ (and its version implemented here in Mathjax, see math.meta.stackexchange.com/questions/5020/…. As for me doing it, lack of parenthesis makes the intention unclear to me, so will not try. At least edit to make the meaning of the formals clear! – kjetil b halvorsen Oct 4 '19 at 9:34
• I guessed that you want formula 2.2a from the linked paper. Can you verify? Also, now push the edit button, look at the $\LaTeX$, and learn! – kjetil b halvorsen Oct 5 '19 at 16:16
• Here is a list of common kernel functions – user20160 Oct 5 '19 at 17:14

$$K$$ is just something you pick as part of the model. The standard choices are indeed Gaussian, Epanechnikov, rectangular, etc. Exactly which properties $$K$$ needs to satisfy will depend on what properties you want, but the general baseline is just that:
• $$\int K(x) \, dx = 1$$ so that the overall estimate integrates to 1;
• usually $$K(x) \ge 0$$ so that, combined with the previous one, the overall estimate is a valid density;
• usually $$K(-x) = K(x)$$ for simplicity.