Timeline for Distribution-free statistics on compact Lie groups
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Apr 1, 2017 at 12:01 | vote | accept | Daniel Miller | ||
| Mar 29, 2017 at 20:38 | comment | added | Asaf | Henry, total variation is the natural distance function (even norm) defined on measures (not necessarily probability). For example it implies convergence in the Wasserstein metric, but indeed, it is very hard for measures to converge in the total variation distance, so usually for various statistical considerations this requirement is relaxed, for example to weak-* convergence in equidistribution questions, or say the wasserstein distance in optimal transport questions and such. The Fourier approach I've mentioned in the abelian compact case would result in weak-* convergence. | |
| Mar 29, 2017 at 19:21 | comment | added | Henry.L | @Asaf So what is the benefit of sup-distance over Wasserstein in this case, altough I think sup distance is also a reasonable choice? Could you explain a bit what you are thinking in your comment above? When I consider it, actually I think Wasserstein is easier to compute. If you suggest sup-distance, then it is simply KS test the OP proposed. And I do not see any superiority of sup-distance in the setup of compact groups. Is there any asymptotic results about sup-distance in this setup? thanks | |
| Mar 29, 2017 at 17:02 | comment | added | Asaf | Daniel - the best choice would be the total variation distance, but this is rather hard to compute and probably stronger than what you have in mind. In many cases you are only interested in weak-* convergence, and limited to proper test function class (say Lip-fcns) a result along your lines is called quantitative equidistribution statement. One possibility is to compute the Fourier-stieltjes coefficients of your empirical measures and try to conclude something, but that usually works best in the abelian case. | |
| S Mar 29, 2017 at 16:35 | history | suggested | Henry.L |
add some relevant tags
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| Mar 29, 2017 at 16:28 | review | Suggested edits | |||
| S Mar 29, 2017 at 16:35 | |||||
| Mar 29, 2017 at 16:25 | comment | added | Henry.L | Feel free to ask more or start a bounty if you are not satisfied. Great question! | |
| Mar 29, 2017 at 15:55 | comment | added | Henry.L | Yes. You are talking about asymptotic distributions, there are quite a few results I can complement in my answer, thanks for the clarification! | |
| Mar 29, 2017 at 15:53 | comment | added | Daniel Miller | I guess the question isn't entirely precise. Given a sequence $\{x_i\}$ of sample points in $G$, drawn from a distribution $\mu$, I'm looking for a function which measures how "close" the empirical measure $\frac{1}{n} \sum_{i=1}^n \delta_{x_i}$ on $G$ is to the true measure $\mu$, and hoping that there is such a function which, when normalized, converges as $n\to \infty$ to a distribution which does not depend on $\mu$. Does that help? | |
| Mar 29, 2017 at 15:50 | answer | added | Henry.L | timeline score: 2 | |
| Mar 29, 2017 at 15:47 | comment | added | Henry.L | What do you mean by "similar" statistics? And do you mean that the asymptotic distribution of the statistics is independent of $F$ or the limiting distribution (mainly moments) of the statistics is independent of $F$. Even in semi-simple Lie groups the notions of limiting distribution and asymptotic distributions do not have to be the same, right? | |
| Mar 27, 2017 at 19:06 | history | asked | Daniel Miller | CC BY-SA 3.0 |