Some more testing examples

Berlinet and Devroye (1994) is an attempt to provide a torture trial of a wide range of KDEs. Appendix contains estimates of each of the 28 densities in the paper, these are as follows:

A histogram.

Providing an example of simple density estimation, a sample of 1000 from the density in question giving a reasonable picture.

An estimated density for a sample of 10000 points using the inbuilt R function density().

R provides a function density that is a highly accurate KDE, this, with a sample of 1000 data items is provided as a reference (referred to as the R estimator in the following).

An estimated density for a sample of 1000 points using BKDE.

An estimated density for a sample of 1000 points using adaptive BKDE.

An estimated density for a sample of 100 points using BKDE.

An estimated density for a sample of 100 points using adaptive BKDE.

Each sub-figure is overlaid with the output from

dberdev(seq(-n,m,0.01),dnum=1)⁵.

which gives the source density.

The histogram and the R density estimator are from samples that are generated for the figure, the bkde and vbkde figures are from the same samples in each case.

Each sample is taken as found and each test is a single sample unless otherwise noted.

Presented here are a few of the estimations where BKDE can be seen to be significantly different to the expected result, from the histogram or from the R estimator.