Kernel Density Estimation for non-parametric

Question

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:

DENSITY ESTIMATION FOR STATISTICS AND DATA ANALYSIS B.W. Silverman School of Mathematics University of Bath, UK.

Answer 1

$K$ 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.

You're not going to be able to write all of this in your code, but you can implement the most common ones and possibly allow for a user-specified one in some way.

@user20160 posted a useful catalog of the most common choices while I was writing this.