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

Abstract

Friedman’s chi-square test is a non-parametric statistical test for r treatments across n trials to assess the null hypothesis that there is no treatment effect. We use Stein’s method with an exchangeable pair coupling to derive a 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 $r∕n\to 0$. From this bound, we deduce a Kolmogorov distance bound that decays to zero under the weaker condition $r^{1∕2}∕n\to 0$.