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Is sum of two negative binomial variables also distributed as negative binomial if they are non-idependent and correlation is $ < 1$? Or not?

den2042
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1 Answers1

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No. For instance if $X$ and $Y$ are independent negative binomials and $Z = X+Y$ (which is negative binomial and has correlation $<1$ with both $X$ and $Y$) then $X+Z=2X + Y$ is not negative binomial. (I'll leave the proof up to you since you changed the question on me.)

spaceisdarkgreen
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  • Great point! Thank you. – den2042 Jun 20 '17 at 22:55
  • Updated the question. – den2042 Jun 20 '17 at 22:59
  • That's not usually how this works, but ok, will update – spaceisdarkgreen Jun 20 '17 at 23:07
  • Updated. $X+Z$ is also a counterexample but lacks the snappy one line proof. – spaceisdarkgreen Jun 20 '17 at 23:13
  • Didn't see you wanted it to be a sum of two, so changed again. At a certain point you have to ask yourself what you're going for. Can you come up with an example of two correlated negative binomial RVs that do sum to a negative binomial? (For that matter, can you come up with a way to construct correlated negative binomials that doesn't feel like 'cheating' to you?) – spaceisdarkgreen Jun 20 '17 at 23:22
  • Thank you very much, spaceisdarkgreen. Very elegant approach. Never thought about the support – den2042 Jun 20 '17 at 23:37
  • @den2042 don't think that works with the new one since the support is still all the integers ... still it would be highly weird if that thing turned out to be a negative binomial. Sorry to be grumpy/lazy and not give a proof. I suggest that there is a relationship between moments that always holds for a neg bin that will fail for $2X+Y.$ – spaceisdarkgreen Jun 20 '17 at 23:45