Timeline for exchangeable normal r.v.s
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 9, 2011 at 23:05 | comment | added | Michael Hardy | (BTW, I still haven't dug Schervish's book out of the library. Otherwise I might have decided what I think of all this by now.) | |
| Sep 8, 2011 at 4:21 | comment | added | Michael Hardy | I'm remembering something else I read some time ago: There are finite exchangeable sequences that cannot be extended to infinite ones. But they can if we allow "negative probabilities", so that expected values become affine combinations but not necessarily convex combinations. The pdf document by Lauritzen mentions that an $n\times n$ covariance matrix of exchangeable r.v.s must have correlations $\ge -1/(n-1)$. But if we want to extend it to an infinitely long exchangeable sequence, then we need non-negative correlations. | |
| Sep 8, 2011 at 4:17 | comment | added | Michael Hardy | I'd forgotten there was such a thing as Hewitt--Savage. | |
| Sep 8, 2011 at 4:10 | comment | added | Michael Hardy | One thing that comes to mind before getting into those results is that for any Borel subset $B$ of the real line, the random variable that equals 1 if $X_n \in B$ and 0 otherwise, is a Bernoulli random variable to which the Bernoulli version of de Finetti's theorem applies. Then we'd have to think about where to go from there..... | |
| Sep 8, 2011 at 4:02 | comment | added | R Hahn | Yes, there are general results to the effect that all exchangeable sequences are mixtures of i.i.d. sequences. See the results describe in these slides: stats.ox.ac.uk/~steffen/teaching/grad/definetti.pdf | |
| Sep 8, 2011 at 3:45 | comment | added | Michael Hardy | The google books link is giving me pages for which it says no preview is available. I can probably locate this in the library tomorrow. | |
| Sep 8, 2011 at 3:43 | comment | added | Michael Hardy | De Finetti's theorem deals with Bernoulli-distributed random variables. Would some some similar result here imply that the exchangeable sequence is a mixture of i.i.d. sequences? | |
| Sep 8, 2011 at 3:39 | comment | added | R Hahn | books.google.com/… | |
| Sep 8, 2011 at 3:29 | history | edited | R Hahn | CC BY-SA 3.0 |
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| Sep 8, 2011 at 3:28 | comment | added | R Hahn | Sorry, the correct term is "transition kernel", and it acts like a conditional probability distribution effectively in this case. | |
| Sep 8, 2011 at 3:23 | comment | added | R Hahn | Mainly I just wanted to advertise the work of Lauritzen (summarized in the section of Schervish that I mention) because it is really beautiful and seems directly relevant. It isn't often that I have the opportunity to bring it up. The missing link is that that work does not talk about prescribing marginal distributions; my thought was that this may be handled using a copula-like construction, but that part is definitely half baked! | |
| Sep 8, 2011 at 3:19 | comment | added | Michael Hardy | What is a "transfer kernel"? Does that mean the conditional probability distribution of the data given the value of the particular statistic? | |
| Sep 8, 2011 at 2:29 | comment | added | Michael Hardy | I knew some of this, and in other parts I'm not sure yet what you're saying (just read it a minute ago). | |
| Sep 8, 2011 at 1:41 | history | edited | R Hahn | CC BY-SA 3.0 |
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| Sep 8, 2011 at 1:36 | history | answered | R Hahn | CC BY-SA 3.0 |