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 $r^{1/2}/n\rightarrow0$.

翻译：Friedman 的 Chi- square 测试是用于 $\ ge2$待遇的非参数统计测试, 用于评估不存在治疗效果的无效假设。 我们使用 Stein 的可互换配方方法, 明确限定 Friedman 统计数据的分布和限制 chi- quare 分布之间的距离, 使用光滑的测试功能进行测量。 我们的绑定是 $\ -1 $, 并且对 $( $) 参数具有最佳依赖性, 即 $( $ ), 约束值在只有 $/ n\ rightroll0 的情况下为零。 我们从这个框中推算出 Kolmogorov 距离, 在较弱的状态下, $r\ 1/ 2} /nrightrow0 下, 折成为零 。