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Bounds for the chi-square approximation of Friedman's statistic by Stein's method

Friedman's chi-square test is a non-parametric statistical test for $r\geq2$ treatments across $n\ge1$ trials to assess the null hypothesis that there is no treatment effect. We use Stein's method with an exchangeable pair coupling to derive an explicit bound on the distance between the distribution of Friedman's statistic and its limiting chi-square distribution, measured using smooth test functions. Our bound is of the optimal order $n^{-1}$, and also has an optimal dependence on the parameter $r$, in that the bound tends to zero if and only if $r/n\rightarrow0$. From this bound, we deduce a Kolmogorov distance bound that decays to zero under the weaker condition $r^{1/2}/n\rightarrow0$.