What do the options of SmoothKernelDistribution do?

Question

The function SmoothKernelDistribution has three options that are not described in too much detail in the Mathematica's help window.

InterpolationPoints: What is interpolated by Mathematica in function SmoothKernelDistribution?

MaxMixtureKernels: As far as my limited knowledge of kernel estimates goes, there will be as many kernels as grid points on my discretized domain. If my data lives on $x$, and I am attempting to approximate probability density function $f(x)$ then I may choose to do so at N evenly-spaced discrete points along $x$, e.g. $x_i, i=1,2,...,N$.

MaxRecursion: What recursion does this option pertain to?

Answer 1

You can click on each of these variables in the help for further explanation.

MaxMixtureKernels: the maximum number of kernels to generate the estimate from. The example in the help file makes this quite clear:

As you can increase the number of kernels (tent poles, if you will) from 10, 15, 25, to 100, the smoother the estimate becomes, at the expense of complexity (more parameters to estimate).

InterpolationPoints: How many points the interpolation function (kernel density estimate) is to be evaluated at.

10 points on the left, 100 on the right. First you fix the number of kernels (consider the previous diagram), then you select where to sample the interpolant.

MaxRecursion: An option for the Plot function to achieve better results in places where more samples are needed. Again, the help file provides some illuminating illustrations:

Here the levels of recursion runs from 0,1,2,4.

Answer 2

@Emre described the options quite well. Worth mentioning here is KernelMixtureDistribution which is a parametric equivalent to SmoothKernelDistribution. The goal of SmoothKernelDistribution is to interpolate the PDF (using linear interpolation) given by KernelMixtureDistribution.

In my work I use KernelMixtureDistribution when speed is less of an issue than quality since it is always a more accurate representation of a kernel density estimator and is capable of handling symbolic inputs. I use SmoothKernelDistribution when I want a quick, numeric and usually visual approximation to some density.

It is also worth pointing out the setting MaxMixtureKernels -> All guarantees that a kernel will be placed at each data point rather than on a uniform grid. This is a good setting to use whenever the number of data points is not astronomically large.