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 ≥ 2 treatments across n ≥ 1 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 → 0. From this bound, we deduce a Kolmogorov distance bound that decays to zero under the weaker condition r 1/2 /n → 0.

Bounds for the chi-square approximation of the power divergence family of statistics

“…Specialising to the case λ = 0 of the likelihood ratio statistic, our O(n −1 ) bound improves on the O(n −1/2 ) rate (and has a better dependence on the other model parameters) of [2] that was given in a more general setting than that of the categorical data considered in this paper. Our results also complement [15] in which Stein's method is used to obtain order O(n −1 ) bounds for the chi-square approximation of Friedman's statistic. In Theorem 3, we provide suboptimal O(n −1/2 ) bounds which may yield smaller numerical bounds for small sample sizes n.…”

“…. , p r via comparison with the asymptotic approximation in (15). We observe that the O(n −1 ) rate of convergence of the upper bounds in Theorem 2 is optimal.…”

“…From (15) this can be seen to be an optimal condition (except perhaps if λ = 2/3). We conjecture that in fact…”

Refined normal approximations for the central and noncentral chi-square distributions and some applications

“…in terms of modified Bessel functions of the first kind, derived from two mean value theorems for definite integrals. Gaunt & Reinert (2021) uses Stein's method to obtain an order n −1 bound on the distributional distance between Friedman's statistics and its limiting chi-square distribution.…”