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If $(X_i,Y_i), i=1,\ldots,n,$ is i.i.d sample from the joint distribution $F$ and there is dependence between the two variables say $R$. Denote the order statistics for the two variables $X_{1:n},\ldots, X_{n:n}$ and $Y_{1:n},\ldots, Y_{n:n}$ respectively.

Now by Renyi's representation, it is possible to show that $X_{n-k:n}$ and $\sum\limits_{i=0} ^{k} X_{n-i:n}-X_{n-k:n}$ are independent, where $k\in{1,\ldots,n-1}$.

I want to check if there is independence or not between $X_{n-k:n}$ and $\sum\limits_{i=0} ^{k} Y_{n-i:n}-Y_{n-k:n}$?

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  • $\begingroup$ I have problems with the index notation. Is the order statistic $1:n,\ldots,n:n$ of $Y$ the same as that for $X$ (i.e. defined by ordering $X$) or is this done separately? In the latter case I see no reason why there should be independence. $\endgroup$
    – Dieter Kadelka
    Commented Apr 5, 2021 at 10:35
  • $\begingroup$ (i) Renyi's representation is only for the case when the $X_i$'s have an exponential distribution. (ii) What are the summation limits in $\sum_i$? (iii) References/proofs for "$X_{n-k:n}$ and $\sum\limits_i X_{n-i:n}-X_{n-k:n}$ are independent"? $\endgroup$
    – Iosif Pinelis
    Commented Apr 5, 2021 at 13:45
  • $\begingroup$ Also, what is $k$? $\endgroup$
    – Iosif Pinelis
    Commented Apr 5, 2021 at 15:25
  • $\begingroup$ @DieterKadelka OS are done separately but there is dependence between the main variables. do you have a way to reason why this dependence is valid?! $\endgroup$
    – Hanan
    Commented Apr 19, 2021 at 16:54
  • $\begingroup$ @IosifPinelis (i) & (iii) yes but there is a way to generally proof that regardless the distribution the independence proof will still hold, check lemma 3.2.3 in extreme value theory an introduction. (ii) the summation is from 0 to k. but regardless the same I need to see if there is dependence between the OS $X_{n-k:n}$ and the difference $Y_{n-i:n}-Y_{n-k:n}$ ! $\endgroup$
    – Hanan
    Commented Apr 19, 2021 at 16:58

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If e.g. the $X_i$'s are independent random variables each uniformly distributed in the interval $[0,1]$, and if (say) $n=2$ and $k=1$, then $U:=X_{n-k:n}$ and $V:=\sum\limits_{i=0}^k(X_{n-i:n}-X_{n-k:n})$ are not independent, because the covariance of $U$ and $V$ is $-1/36\ne0$.

If you now take $Y_i:=X_i$ for all $i$, then $X_{n-k:n}$ and $\sum\limits_{i=0}^k(Y_{n-i:n}-Y_{n-k:n})$ will not be independent.

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