Refinement on the convergence of one family of goodness-of-fit statistics to chi-squared distribution

Ulyanov, Vladimir V.; Zubov, V. N.

doi:10.32917/hmj/1237392382

Cited by 17 publications

(12 citation statements)

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“…For general results and application to Pearson's statistic see [24], in which a O(n −1 ) Kolmogorov distance bound between Pearson's statistic and the χ 2 (r−1) distribution was obtained for the case of r ≥ 6 cell classifications, improving a result of [43]. Also, [2,42] have used Edgeworth expansions to study the rate of convergence of the more general power divergence family of statistics constructed from the multinomial distribution of degree r (which includes the Pearson, log-likelihood ratio and Freeman-Tukey statistics as special cases) to their χ 2 (r−1) limits. Using Stein's method [40], a O(n −1 ) bound for the rate of convergence for Pearson's statistic for r ≥ 2 cell classifications was obtained by [18].…”

Section: Introduction 1friedman's Statistic and Main Resultsmentioning

confidence: 99%

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

Gaunt¹,

Reinert²

2021

Preprint

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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.

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Section: Introduction 1friedman's Statistic and Main Resultsmentioning

confidence: 99%

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

Gaunt¹,

Reinert²

2021

Preprint

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“…to the chi-square distribution. Some lapses in the expansions of Siotani & Fujikoshi (1984) and Read (1984), regarding the rate of convergence of the chi-square approximation, were pointed out and fixed in Ulyanov & Zubov (2009) (see also Prokhorov & Ulyanov (2013)).…”

Section: Other Potential Applicationsmentioning

confidence: 99%

A precise local limit theorem for the multinomial distribution and some applications

Ouimet

2021

Journal of Statistical Planning and Inference

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In Siotani & Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N −1 . In this paper, we give an alternative (conceptually simpler) proof based on Stirling's formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of R d . Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter ( 2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.

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“…The approach of Götze and Ulyanov [16] was further developed in [24] for the power divergence family of statistics and then refined in [1,2]:…”

Section: Phi-divergence Test Statisticmentioning

confidence: 99%

Inference via Randomized Test Statistics

Puchkin¹,

Ulyanov²

2021

Preprint

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We show that external randomization may enforce the convergence of test statistics to their limiting distributions in particular cases. This results in a sharper inference. Our approach is based on a central limit theorem for weighted sums. We apply our method to a family of rank-based test statistics and a family of phi-divergence test statistics and prove that, with overwhelming probability with respect to the external randomization, the randomized statistics converge at the rate O(1/n) (up to some logarithmic factors) to the limiting chi-square distribution in Kolmogorov metric.

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Refinement on the convergence of one family of goodness-of-fit statistics to chi-squared distribution

Cited by 17 publications

References 0 publications

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

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

A precise local limit theorem for the multinomial distribution and some applications

Inference via Randomized Test Statistics

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