Timeline for q-Means and the mode of a distribution
Current License: CC BY-SA 4.0
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| Oct 10, 2019 at 8:05 | history | edited | YCor | CC BY-SA 4.0 |
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| Oct 9, 2019 at 14:44 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Oct 9, 2019 at 14:14 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jul 1, 2018 at 10:33 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jun 30, 2018 at 14:43 | vote | accept | Maurizio Barbato | ||
| Jun 30, 2018 at 14:43 | answer | added | Maurizio Barbato | timeline score: 1 | |
| Jun 29, 2018 at 14:57 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jun 28, 2018 at 18:11 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jun 28, 2018 at 17:59 | comment | added | Maurizio Barbato | @fedja To compute $F_q(y)$ for the counterexample you have in mind is an easy (even though boring) task. But to study the behavior of $S_q$ while $q$ approaches 0 seems not trivial at all: of course we could approach this last task numerically, but I am looking for a rigorously examined counterexample, not a tentative one. Anyway, generally speaking, the more I think of this problem, the more it is clear to me that there is no compelling reason for which the conjecture should hold true. | |
| Jun 28, 2018 at 9:31 | comment | added | fedja | @MaurizioBarbato The counterexample I suggested is unimodal. Check it. | |
| Jun 27, 2018 at 7:57 | comment | added | Maurizio Barbato | @fedja Do you think you can find a counter-example in which the distribution is strongly unimodal, that is $f$ is non-decreasing on $(-\infty,x_0]$ and non-increasing on $[x_0, \infty)$? Henry said to suspect that in this case the conjecture could be true, but I am not convinced at all. | |
| Jun 26, 2018 at 3:21 | comment | added | usul | By the way, is it true that the mode is the limit of the maximizer as $q \to -\infty$? (maybe under some conditions on $f$) | |
| Jun 26, 2018 at 2:38 | comment | added | fedja | It looks like if it converges to anything at all, that anything should be a minimizer of $\int\log|y-x|f(x)\,dx$, which certainly does not need to be the mode even in the unimodal case. So I would bet on the claim being false. If nobody comes to a definitive conclusion in the next few days, I'll try to make an accurate computation. The suspicious case is when $f$ is essentially the characteristic function of $[-1,1]$ with a small upward bump near $1$. | |
| Jun 25, 2018 at 22:55 | answer | added | Henry | timeline score: 2 | |
| Jun 25, 2018 at 15:19 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jun 25, 2018 at 13:41 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jun 25, 2018 at 13:23 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jun 24, 2018 at 19:42 | history | edited | Maurizio Barbato |
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| Jun 24, 2018 at 13:01 | history | edited | Maurizio Barbato | CC BY-SA 4.0 |
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| Jun 24, 2018 at 12:51 | history | asked | Maurizio Barbato | CC BY-SA 4.0 |