Rebirth

Between trying to get things tied up at work before leaving for Ithaca and the trip itself, I haven't had much to report the past week. Thus, there have been no reports.

However, one of the advantages of taking a break is that you tend to think of the bigger picture a bit. I'm starting to think this whole bootstrap business has some promise. It's an active area, so it will be viewed as relevant and nobody seems to be terribly concerned that all the existing methods fail (or, at least have not been proven correct) when higher moments are absent. So, one might conclude that getting some results in the absence of higher moments would be a good thing.

Here are a few of the questions that might be worth answering (I'll have to check to see if any of them have been):

1. How can you mix bootstrapping with stratification?
2. How to account for mixture distributions where the mixture has temporal correlation?
3. More detailed results using BMH with mixture/hierarchical models. In particular, does a block sampling strategy have the same asymptotic properties, even in the presence of block correlation (I'm sure it does, but the proof could be messy)?
4. What happens to all these things when the variance is unbounded?
5. What happens when we mix the correlation-maximizing strategy of D-BESt with methods designed to account for correlation?
6. What if we replaced the pseudo-random number generator with a chaotic function in these methods?

That last one is a bit off-topic, but it occurred to me that random number generators are really just chaos functions by another name. Maybe it would be simpler/more efficient/more effective to intentionally pick a chaos function that covers the the sample space non-uniformly. Sometimes what we want isn't so much randomness, but non-periodic coverage. Non-linear differential equations are good at that, particularly when you go to higher dimensions.

Also, I still have my eye on the Blind Multi-Stage Bootstrap that I wrote about a couple weeks ago. A couple additional thoughts on that one:

1. Does it make sense to update the BMH distribution with each sample based on what we've observed so far? If so, what sort of prior do you use to control that?
2. What if it's the entire model that you're not sure about? That is, rather than a few competing models, is there a way to be entirely agnostic about the model? We actually had a colloquium on this last year which looked into Dirichlet models. I think I'd at least like to read up on that.