In general, the AMISE still depends of the true underlying density (which of course we don't have!) and so we need to estimate the AMISE from our data as well. This means that the chosen bandwidth is an estimate of an asymptotic approximation. It now sounds as if it's too far away from the true optimal value but it turns out that this particular choice of bandwidth recovers all the important features whilst maintaining smoothness.
The optimal value of the bandwidth for our dataset is about 0.25. From the optimally smoothed kernel density estimate, there are two modes. As these are the log of aircraft wing span, it means that there were a group of smaller, lighter planes built, and these are clustered around 2.5 (which is about 12 m). Whereas the larger planes, maybe using jet engines as these used on a commercial scale from about the 1960s, are grouped around 3.5 (about 33 m).
0 && p_w_picpath.height>0){if(p_w_picpath.width>=700){this.width=700;this.height=p_w_picpath.height*700/p_w_picpath.width;}}" align=center>
The properties of kernel density estimators are, as compared to histograms:
- smooth
- no end points
- depend on bandwidth
This has been a quick introduction to kernel density estimation. The current state of research is that most of the issues concerning one-dimensional problems have been resolved. The next stage is then to extend these ideas to the multi-dimensional case where much less research has been done. This is due to that there are the orientation of multi-dimensional kernels has a large effect on the resulting density estimate (which has no counterpart in one-dimensional kernels). I am currently looking for reliable methods for bandwidth selection for multivariate kernels. Some progress that I have made in plug-in methods is
here. However this page is more technical and uses equations!
These notes are an edited version of a seminar given by the author (Tarn Duong) on 24 May 2001 as part of the Weatherburn Lecture Series for the Department of Mathematics and Statistics, at the University of Western Australia. Please feel free to contact the author at duongt(at)maths.(dot)uwa(dot)edu(dot)au if you have any questions.