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To test that a sample $X_1,\ldots,X_n$ are i.i.d against that the distributions of $X_i$ are stochastically increasing in $i$, how to find the distribution of the test statistic and the critical value for large $n$?

I suppose that the U-statistic, $U=\frac{1}{n \choose 2}\sum_{i<j}1\{X_j-X_i>0\}$. But how do I find the critical value?

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    $\begingroup$ The Mann-Kendall trend test uses the Kendall correlation of the variable with time (note that $\sum_{i<j}1\{X_j-X_i>0\}$ is simply a count of increasing - i.e. 'concordant with time' - pairs). Tables of the Kendall correlation are available, come built-in to many stats packages, and the normal approximation kicks in very quickly (note that when doing the normal approximation in a Mann-Kendall trend test, a continuity correction often seems to be applied). Which do you seek? $\endgroup$ – Glen_b Oct 8 '18 at 0:20
  • $\begingroup$ I'd like to know about the normal approximation with continuity correction. Thanks! $\endgroup$ – Blain Waan Oct 8 '18 at 17:20

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