# Critical value for Mann's test against trend

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, $$U=\frac{1}{n \choose 2}\sum_{i0\}$$. But how do I find the critical value?

• 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? – Glen_b Oct 8 '18 at 0:20
• I'd like to know about the normal approximation with continuity correction. Thanks! – Blain Waan Oct 8 '18 at 17:20