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**A framework for Monte Carlo based multiple testing.**
*(English)*
Zbl 1373.62389

Summary: We are concerned with a situation in which we would like to test multiple hypotheses with tests whose \(p\)-values cannot be computed explicitly but can be approximated using Monte Carlo simulation. This scenario occurs widely in practice. We are interested in obtaining the same rejections and non-rejections as the ones obtained if the \(p\)-values for all hypotheses had been available. The present article introduces a framework for this scenario by providing a generic algorithm for a general multiple testing procedure. We establish conditions that guarantee that the rejections and non-rejections obtained through Monte Carlo simulations are identical to the ones obtained with the \(p\)-values. Our framework is applicable to a general class of step-up and step-down procedures, which includes many established multiple testing corrections such as the ones of Bonferroni, Holm, Sidak, Hochberg or Benjamini-Hochberg. Moreover, we show how to use our framework to improve algorithms available in the literature in such a way as to yield theoretical guarantees on their results. These modifications can easily be implemented in practice and lead to a particular way of reporting multiple testing results as three sets together with an error bound on their correctness, demonstrated exemplarily using a real biological dataset.

### MSC:

62J15 | Paired and multiple comparisons; multiple testing |

65C05 | Monte Carlo methods |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

### Keywords:

algorithm; framework; hypothesis testing; Monte Carlo; multiple testing procedure; \(p\)-value
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\textit{A. Gandy} and \textit{G. Hahn}, Scand. J. Stat. 43, No. 4, 1046--1063 (2016; Zbl 1373.62389)

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