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We introduce a simple tool to control for false discoveries and identifyindividual signals in scenarios involving many tests, dependent teststatistics, and potentially sparse signals. The tool applies the Cauchycombination test recursively on a sequence of expanding subsets of $p$-valuesand is referred to as the sequential Cauchy combination test. While theoriginal Cauchy combination test aims to make a global statement about a set ofnull hypotheses by summing transformed $p$-values, our sequential versiondetermines which $p$-values trigger the rejection of the global null. Thesequential test achieves strong familywise error rate control, exhibits lessconservatism compared to existing controlling procedures when dealing withdependent test statistics, and provides a power boost. As illustrations, werevisit two well-known large-scale multiple testing problems in finance forwhich the test statistics have either serial dependence or cross-sectionaldependence, namely monitoring drift bursts in asset prices and searching forassets with a nonzero alpha. In both applications, the sequential Cauchycombination test proves to be a preferable alternative. It overcomes many ofthe drawbacks inherent to inequality-based controlling procedures, extremevalue approaches, resampling and screening methods, and it improves the powerin simulations, leading to distinct empirical outcomes.

Sequential Cauchy Combination Test for Multiple Testing Problems with Financial Applications