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

Understand

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