Inference via randomized test statistics

Annales de l'institut Henri Poincare (B) Probability and Statistics. 2023. Vol. 59. No. 3. P. 1508–1529.

Nikita Puchkin, Vladimir Ulyanov

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.

Keywords: central limit theorem, центральная предельная теорема, weighted sums, Phi-divergence test statistics, Power divergence test statistics, Rank-based test statistics, Взвешенные суммы, Ранговые статистики, Дивергенции Кресси-Рида