Timeline for Kendall's tau and independence
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2012 at 18:54 | history | notice removed | CommunityBot | ||
| Apr 6, 2012 at 18:54 | history | bounty ended | CommunityBot | ||
| Apr 1, 2012 at 4:37 | vote | accept | user7064 | ||
| Mar 31, 2012 at 20:30 | history | edited | chl |
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| Mar 31, 2012 at 17:38 | history | tweeted | twitter.com/#!/StackStats/status/186145568232452096 | ||
| Mar 31, 2012 at 16:39 | answer | added | cardinal | timeline score: 18 | |
| Mar 30, 2012 at 13:59 | comment | added | whuber♦ | Thanks, Cardinal: that is the clarification I was attempting to elicit. | |
| Mar 30, 2012 at 4:24 | comment | added | user7064 | @cardinal: Yes, indeed :-) | |
| Mar 29, 2012 at 23:55 | comment | added | cardinal | @whuber: I think you might be thinking of the sample version of Kendall's $\tau$. As a population quantity, this is defined as $\tau = \mathbb P((X-X')(Y-Y') > 0) - \mathbb P((X-X')(Y-Y')<0)$ where $(X,Y)$ and $(X',Y')$ are iid bivariate vectors with continuous distributions. From this definition, it is clear that if $X$ and $Y$ are independent, then $\tau = 0$. The OP seems to be asking for a characterization of the reverse implication. | |
| Mar 29, 2012 at 17:51 | comment | added | whuber♦ | Normally, $\tau$ is a sample statistic and is not considered a parameter of a joint distribution. Therefore, when $X$ and $Y$ are independent with joint distribution $F$ and $(x_i,y_i)$ is an iid sample from $(X,Y)$, $\tau$ is a function of this sample and we can say $\tau$ is $0$ only in expectation: $\mathbb{E}_F[\tau]=0$. In particular, $\tau=0$ does not even imply lack of correlation of $X$ and $Y$: it merely is statistical evidence of lack of correlation. If these considerations lead you to want to modify your question, then please go ahead and edit it. | |
| Mar 29, 2012 at 17:33 | history | notice added | user7064 | Canonical answer required | |
| Mar 29, 2012 at 17:33 | history | bounty started | user7064 | ||
| Mar 28, 2012 at 17:19 | comment | added | Cyan | Check out distance correlation. | |
| Mar 27, 2012 at 15:18 | comment | added | whuber♦ | This question needs disambiguation. Does it ask whether (a) there are conditions on $X$ and $Y$ that assure $\tau=0$ implies independence of $X$ and $Y$ or (b) there are alternative statistics other than $\tau$ whose vanishing implies independence of $X$ and $Y$, not matter how $(X,Y)$ are distributed? | |
| Mar 27, 2012 at 15:09 | history | edited | user7064 | CC BY-SA 3.0 |
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| Mar 27, 2012 at 15:04 | comment | added | cardinal | There are quite a few, actually. Some come from statistics and other come from closely allied fields, like information theory. Specifying your model in a little more detail can yield more precise suggestions. | |
| Mar 27, 2012 at 13:21 | history | asked | user7064 | CC BY-SA 3.0 |