Suppose we have $N$ independent random variables $X_1, \cdots, X_N$ drawn from $f_1 > \cdots > f_N$ where $f_i > f_j$ indicates that $f_i$ and $f_j$ satisfy the monotone likelihood ratio property (MLRP). That is, $\forall a<b$ $$ \frac{f_j (a)}{f_j (b)} > \frac{f_i(a)}{f_i(b)}$$.

Now suppose that for every realization of the random variables $x_1, \cdots, x_n$ we construct a statistic $O_i$ containing the rank order position of each variable. That is, if we have $N=4$ $x_1= 100$ $x_2= 5$ $x_3= 99$ $x_4 = 2$ then $o_1 = 1$ $o_2 = 3$ $o_3 = 2$ $o_4 = 4$.

Is it true that for $i<j$ $O_i>O_j$ in the MLRP sense? I have a feeling that if it's true it must be known in the stats/probability literature.

Thanks a lot.


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