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I'm trying to figure out how variance of Kendall's tau statistic is determined in the presence of ties. From Yue et al. 2002:

The variance of $S$ is given by:

$E(S) = \frac{n(n-1)(2n+5)-\sum_{m=1}^{n}t_{m}m(m-1)(2m+5)}{18}$

where $t_m$ is the number of ties of extent $m$.

What does "of extent $m$" mean?

Yue, Sheng, et al. "The influence of autocorrelation on the ability to detect trend in hydrological series." Hydrological processes 16.9 (2002): 1807-1829.

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  • $\begingroup$ Please give a full reference so we have some chance of checking the context. "Yue et al" and a year isn't much to go on. Is this Yue, S., Pilon, P. & Cavadias, G. (2002), "Power of the Mann-Kendall and Spearman’s rho tests for detecting monotonic trends in hydrological series.", J. Hydrol. 259 , 254–271 or something else? $\endgroup$ – Glen_b -Reinstate Monica Jul 3 '18 at 2:44
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Presumably $m$ represents the number of values that are tied at some particular number. For example, if your sample were sorted into increasing order and had $(1.9, 2.0, 2.0, 3.1, 3.5, 3.5, 3.5, 3.8, 4.3, 4.3, 5.0)$ there's a pair of tied values at each of $2.0$ and $4.3$ and a triple at $3.5$ (a pair of them means $m=2$, a triple means $m=3$) while $t_2$ represents the number of times you get $m=2$ across the whole sample (the number of values with ties where there are just two values tied). So here $t_2=2$ and $t_3=1$ (while $t_4=0$ because there were no values in our sample where 4 observations were all tied together).

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