Timeline for Extension of subcopulas to copulas
Current License: CC BY-SA 4.0
17 events
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| Jan 13, 2022 at 22:01 | comment | added | Anthony Quas | In higher dimensions, Area is Lebesgue measure. If you substitute $[0,1]^2$, you see the formula gives 1. You can correct the formula as you have done, but I find that unpleasant. I would prefer to restrict the sum those A that do not lie completely in Z (or just define the summand to be 0 in those cases). | |
| Jan 13, 2022 at 15:00 | comment | added | Star | Q4: to avoid "$\frac{0}{0}$", is your formula equivalent to $$\lambda(S)=\sum_{A \text{ atomic}} \tilde{\mu}(S\cap A)\frac{\text{Area}(A\cap S\cap Z^c)}{\Big[\text{Area}(A\cap Z^c)\times 1\{\text{$A$ is not contained in single zero-measure box}\} +1\{\text{$A$ is contained in single zero-measure box} \}\Big]}$$ | |
| Jan 13, 2022 at 14:00 | comment | added | Star | Thanks! Q1 (second half): what is "Area" in dimension $>2$? Is it "Lebesgue measure"? Q2: I'm not understanding this step: how do you know that 𝜆 takes values in [0,1] and that is uniform (so that we can call it a copula)? | |
| Jan 12, 2022 at 22:40 | comment | added | Anthony Quas | Q1 (first half): So I am not distinguishing between copulas and measures because they're really the same thing. $S$ can be any (measurable) subset. Q2,3: To see that $\lambda$ is an extension, it suffices to check on atomic rectangles. It just is a copula because it's a measure. Q4: if the denominator is 0, then $\bar\mu(S\cap A)$ is required to be 0 and I just ignore the term. Q1 (second half): I use Area rather than $\bar\mu$ because we don't know anything about $\bar\mu$. Maybe $\bar\mu$ sneakily puts zero mass on $A\cap Z^c$ for some $A$. | |
| Jan 12, 2022 at 19:02 | comment | added | Star | Fourth question: in the denominator of the equation, $\text{Area}(A\cap Z^c)$ could be zero if $A\cap Z^c=\emptyset$. This is the case when the WHOLE $A$ lies in a SINGLE zero-measure box, which can happen. Therefore, I wonder whether perhaps you meant $$\sum_{A\text{ atomic}: \text{$A$ intersects $S$ and is not contained in a single zero box}} \bar{\mu}(S\cap A)\frac{\bar{\mu}(A\cap S\cap Z^c)}{\bar{\mu}(A\cap Z^c)}+\underbrace{\sum_{A\text{ atomic}: \text{$A$ intersects $S$ and is contained in a single zero box}} \bar{\mu}(S\cap A)}_{=0}$$ | |
| Jan 12, 2022 at 17:40 | comment | added | Star | Third question: how do we know that $\lambda$ is an extension of $\bar{\mu}$? | |
| Jan 12, 2022 at 17:12 | comment | added | Star | Also, why, to see that $\lambda$ is a copula, it suffices to check that it agrees with $\bar{\mu}$ on the atomic rectangles? | |
| Jan 12, 2022 at 17:04 | comment | added | Star | I was reviewing your answer and I'm having doubts on the notation of the second part, where you add details. In particular, I'm referring to $$\lambda(S)=\sum_{A \text{ atomic}} \bar{\mu}(S\cap A) \frac{\text{Area}(A\cap S \cap Z^c)}{\text{Area}(A\cap Z^c)}$$ What is $S$? My understanding is that $S$ is a subset (nor necessarily a rectangle) of $[0,1]^2$ (if, for instance, we are in dimension 2). Therefore, I believe that $\lambda$ and $\bar{\mu}$ are not really copulas, but measures associated with copulas. Is this correct? If yes, why do you write "Area", instead of $\bar{\mu}$? | |
| Dec 18, 2021 at 21:18 | history | bounty ended | Star | ||
| Dec 18, 2021 at 21:18 | vote | accept | Star | ||
| Dec 18, 2021 at 5:44 | history | edited | Anthony Quas | CC BY-SA 4.0 |
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| Dec 15, 2021 at 7:12 | history | edited | Anthony Quas | CC BY-SA 4.0 |
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| Dec 15, 2021 at 4:55 | history | edited | Anthony Quas | CC BY-SA 4.0 |
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| Dec 15, 2021 at 4:37 | comment | added | Anthony Quas | Yes. I agree with your interpretation of my answer. | |
| Dec 14, 2021 at 17:28 | comment | added | Star | Also, in the link I post at the end of this comment, the authors seem to describe a "swapping" algorithm of masses which resembles your suggestion. However, differently from my case, they do not have to maintain the mass at the points in $\mathcal{S}$ fixed. books.google.co.uk/… | |
| Dec 14, 2021 at 14:14 | comment | added | Star | Thanks, I really appreciate this, although things are not completely clear to me. I have added the questions that I have at the bottom of my answer because they were too involved to be written in comments. | |
| Dec 14, 2021 at 7:48 | history | answered | Anthony Quas | CC BY-SA 4.0 |