Nonparametric Methods – Why Kernel Density Estimation Remains Nonparametric with Parametrized Kernel

distributionskernel-smoothingnonparametric

I am new to kernel density estimation (KDE), but I want to learn about it to help me calculate probabilities of outcomes in sequencing data. I watched this https://www.youtube.com/watch?v=QSNN0no4dSI as my first introduction to the subject.

As the lecturer was going over different kernels, I realized it confused me that KDE is considered non-parametric even when the kernel was being locally parameterized by points within a bandwidth.

Are the standard deviation and arithmetic mean not parameters of KDE when the kernel is the normal distribution?

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Doing a little more reading, I have seen different definitions that change my perspective of non-parametric models. I believed that a non-parametric model/distribution must entirely lack parameters, but some report that non-parametric statistics may or may not have parameters. What distinguishes non-parametric statistics from parametric statistics is that they do not have a fixed or a priori distribution or model structure.

To answer the question, KDE with a normal kernel does not assume that the resulting distribution will have a particular shape, and does not assume a model structure (Ex. the number of parameters). Thus, KDE with a normal kernel is non-parametric.

Related Solutions

Solved – FFT for speeding up kernel density estimation

Recently, the problem was successfully solved. Please, see https://arxiv.org/abs/1508.02766 for details. Similar problem, relating the unconstrained bandwidth selection, was presented here https://arxiv.org/abs/1511.07482

Solved – np package kernel density estimation with Epanechnikov kernel

EDIT

This is explained in the FAQ:

I use plot() (npplot()) to plot, say, a density and the resulting plot looks like an inverted density rather than a density

This can occur when the datadriven bandwidth is dramatically undersmoothed. Data-driven (i.e., automatic) bandwidth selection procedures are not guaranteed always to produce good results due to perhaps the presence of outliers or the rounding/discretization of continuous data, among others. By default, npplot() takes the two extremes of the data (minimum, maximum i.e., actual data points) then creates an equally spaced grid of evaluation data (i.e., not actual data points in general) and computes the density for these points. Since the bandwidth is extremely small, the density estimate at these evaluation points is correctly zero, while those for the sample realizations (in this case only two, the min and max) are non-zero, hence we get two peaks at the edges of the plot and a flat bowl equal to zero everywhere else. This can also happen when your data is heavily discretized and you treat it as continuous. In such cases, treating the data as ordered may result in more sensible estimates

As suggested treating the data as ordered, works:

blep<-npudensbw(~ordered(geyser$waiting), 
                bwmethod="cv.ls", ckertype="epanechnikov", ckerorder=2)

It also succeeds with higher kernel orders, such as with ckerorder=4 in this example:

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Related Solutions

Solved – FFT for speeding up kernel density estimation

Solved – np package kernel density estimation with Epanechnikov kernel

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