There is a slight flaw in the kernel-smoothing approach to getting 2-D
confidence regions. The confidence regions are conservative because the
variance of the smoothing kernel is added to the variation in the data.
In fact the confidence regions grow in size as the bandwidth increases.
If you play about with the mccontour() function you will see this.
Ideally the confidence regions should stay the same size, but become
more smooth in this limit.
So I have two questions
1) Is this a serious problem? Bear in mind that with MCMC one can always
generate more data and keep the same (small) kernel size.
2) How would you go about getting a density estimate that gave
unbiased confidence regions? I'm wondering if a Bayesian desnsity
estimate (in which the log density is modelled by an autoregressive
process) would have this property.
I know this isn't a general discussion forum, but it is an interesting
problem. Send your thoughts to me and I will summarize.
Martyn
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