Imagine I want to statistically characterise a set of converging points and still get an idea of the converging properties or shape of such set, for example

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The values of the mean or variance of the $y$ coordinates don't really tell me anything specific about the converging shape of such set and I wouldn't be able to, just by looking at such values, distinguish the previous set from something like

enter image description here

I could, however, artificially build something depending on the $x$ coordinates (the variance, for example). Naturally, in the first set, for points where $x>L$, we expect the variance of the $y$ coordinates to decrease as $L$ increases, but this seems too much, so I was wondering if there is a neater way of statistically characterising such sets and being able to distinguish both previous sets. Any ideas?

  • $\begingroup$ Does $\mathrm{Cov}(X,|Y|)$ work? The first one will be negative and the second one should be close to 0. $\endgroup$
    – Q9y5
    Dec 3 '21 at 12:35
  • $\begingroup$ @Q9y5 but would I be able to distinguish it from a diverging set of points, for example? $\endgroup$
    – sam wolfe
    Dec 3 '21 at 15:25
  • $\begingroup$ For diverging set, I guess $\mathrm{Cov}(X,|Y|)$ should be positive. You can do a simulation to verify it. $\endgroup$
    – Q9y5
    Dec 3 '21 at 15:30
  • $\begingroup$ Thanks, will do $\endgroup$
    – sam wolfe
    Dec 3 '21 at 15:37
  • $\begingroup$ Make the histogram and match it to one of the continuous distributions. It looks like the pdf will be exponential decay like beta distribution or chi or something. It’s not normal $\endgroup$
    – Joe Banden
    Dec 5 '21 at 2:07

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