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 ≥ 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 r1/2/n → 0.

[1]  G. Reinert,et al.  Stein's method and the zero bias transformation with application to simple random sampling , 1997, math/0510619.

[2]  C. Stein Approximate computation of expectations , 1986 .

[3]  S. Chatterjee A NEW METHOD OF NORMAL APPROXIMATION , 2006, math/0611213.

[4]  F. Götze On the Rate of Convergence in the Multivariate CLT , 1991 .

[5]  V. Ulyanov,et al.  Refinement on the convergence of one family of goodness-of-fit statistics to chi-squared distribution , 2009 .

[6]  C. Stein A bound for the error in the normal approximation to the distribution of a sum of dependent random variables , 1972 .

[7]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[8]  Robert E. Gaunt Rates of convergence of variance-gamma approximations via Stein's method , 2013 .

[9]  G. Reinert,et al.  Stein's method for comparison of univariate distributions , 2014, 1408.2998.

[10]  Robert E. Gaunt,et al.  Chi-square approximation by Stein's method with application to Pearson's statistic , 2015, 1507.01707.

[11]  Elizabeth S. Meckes,et al.  On Stein's method for multivariate normal approximation , 2009, 0902.0333.

[12]  The Gamma Stein equation and noncentral de Jong theorems , 2016, Bernoulli.

[13]  Murat A. Erdogdu,et al.  Flexible results for quadratic forms with applications to variance components estimation , 2015, 1509.04388.

[14]  Exact norms of a Stein-type operator and associated stochastic orderings , 2003 .

[15]  Terry L King A Guide to Chi-Squared Testing , 1997 .

[16]  Wei-Liem Loh Stein's Method and Multinomial Approximation , 1992 .

[17]  Xiao Fang,et al.  High-Order Steady-State Diffusion Approximations , 2020, Operations Research.

[18]  G. Reinert,et al.  Bounds for the asymptotic distribution of the likelihood ratio , 2018, The Annals of Applied Probability.

[19]  J. Marcinkiewicz Sur les fonctions indépendantes , 1938 .

[20]  Robert E. Gaunt Bounds for the chi-square approximation of the power divergence family of statistics , 2021, Journal of Applied Probability.

[21]  J. K. Yarnold Asymptotic Approximations for the Probability that a Sum of Lattice Random Vectors Lies in a Convex Set , 1972 .

[22]  M. Fathi Higher-order Stein kernels for Gaussian approximation , 2018, Studia Mathematica.

[23]  On approximating some statistics of goodness-of-fit tests in the case of three-dimensional discrete data , 2011 .

[24]  A. Barbour Stein's method for diffusion approximations , 1990 .

[25]  Robert E. Gaunt Stein’s method for functions of multivariate normal random variables , 2015, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[26]  Martin Raič,et al.  Normal Approximation by Stein ’ s Method , 2003 .

[27]  Zero Biasing in One and Higher Dimensions, and Applications † ‡ , 2004 .

[28]  Yosef Rinott,et al.  Multivariate normal approximations by Stein's method and size bias couplings , 1996 .

[29]  E. Peköz,et al.  New rates for exponential approximation and the theorems of Rényi and Yaglom , 2011, The Annals of Probability.

[30]  S. Chatterjee,et al.  MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS , 2007, math/0701464.

[31]  Gesine Reinert,et al.  Stein's density method for multivariate continuous distributions , 2021 .

[32]  M. Friedman The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance , 1937 .

[33]  P. Diaconis,et al.  Closed Form Summation for Classical Distributions: Variations on Theme of De Moivre , 1991 .

[34]  Jason Fulman,et al.  Exponential Approximation by Stein's Method and Spectral Graph Theory , 2006 .

[35]  G. Reinert,et al.  Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition , 2007, 0711.1082.

[36]  I. Shevtsova On the absolute constants in the Berry-Esseen-type inequalities , 2011, 1111.6554.