Asymptotic error of Bonferroni procedure under weak dependence via Chen–Stein method

In this paper, we are interested in the limiting distribution of the number of false positives when using Bonferroni adjustment for large-scale multiple testing, as the number of hypotheses grows to infinity. It is proven in the literature that the distribution converges to a Poisson distribution, if the statistics are positively equi-correlated normal but nearly independent. In this paper, we provide an alternative proof using the Chen–Stein method. Unlike existing works, our proof provides a rate of convergence of the number of false positives to its asymptotic distribution, which we also confirm numerically. In addition, we show that our results are applicable to a more generalized setting beyond the equi-correlation assumption.