Talk:Goodman and Kruskal's gamma

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Distribution of the test statistics for $\gamma$

Latest comment: 8 hours ago1 comment1 person in discussion

Critical values for the gamma statistic are sometimes found by using an approximation, whereby a transformed value, t of the statistic is referred to Student t distribution, where^{[citation needed]}

t\approx G{\sqrt {\frac {N_{s}+N_{d}}{n(1-G^{2})}}}

I think the test statistics is not $t$ distributed, but asymptotical $N(0;1)$ . After all $G{\sqrt {\frac {N_{s}+N_{d}}{n(1-G^{2})}}}$ is a discrete test statistics that can be approximated by a standard normal if $n$ is large enough. If $n$ is small then an exact test would be required.

Additionally, I did not find any paper stating that $t$ is $t$ -distributed. Even Goodman and Kruskal only discuss asymptotic variances. Also, all the software implementations I looked at use the z-score for the test."