As I mentioned in my post discusssing
Kernel Density Estimators for the Dynamic Trading Risk Factor, one nice
property of histogramming is that the sampling errors for the
kernel density estimate that the histogram represents and well understood and
straightforward to compute. Computing the sampling distribution for the
estimator is considerably more complicated for kernel density estimators.
The above chart was prepared in
Mathematica
.
On the laptop I have Mathematica 6 installed on, the k.d.e. chart
takes about 30 seconds to run (Mathematica is a symbolic computation
system and, as such, will always execute on the slower side), but the
computation of the error bands took several hours — I set the job up at
midnight and looked at the results over breakfast — and the resulting
data is not particularly profound!
The expression for the mean square error of the point estimator
is
(x)-(K_h*f)^2(x)}{n}+\left\{(K_h*f)(x)-f(x)\right\}^2)
.
However, this expression is written in terms of the true population density
and, since the entire premise of kernel density estimation is to estimate the
density, we clearly do not know that. As the procedure is
relatively efficient, when compared to histogramming, and an
unbiased estimator, we can assume that the density estimate
converges in expectation to the true population density. This
justifies the step of replacing f(x)
with its estimator in the above expression, which was what I did to compute the
error bands.
I started this analysis to look at whether there was evidence for clustering
of factor returns in the region of 2%, but am not seeing that in these
procedures with a standard choice of bandwidth. The book I've been
working from,
Ward and Jones's Kernel Smoothing (Monographs on Statistics and Applied
Probability)
is silent on the sampling distributions for the estimators. Although I drew
error bands on the plot, I don't actually know the probability that the true
density estimate lies within those bands at a given point. We can reach for the
Central Limit Theorem, and suggest that it is in the region of
68% of the probability mass — but that is truthfully just a guess. I think I have
to go back to histogramming, with an arbitary bin-width, to assess whether the
clustering is statistically significant.