On the Simes test under dependence
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In 1986, R. J. Simes proposed a modified Bonferroni test procedure for testing an overall null hypothesis in multiple testing problems, nowadays referred to as the Simes test. The paper of Simes may be considered as a basic step in the development of many new test procedures and new error rate criteria as for example control of the false discovery rate. A key issue is the validity of the Simes test and the underlying Simes inequality under dependence. Although it has been proved that the Simes inequality is valid under suitable assumptions on dependence structures, important cases are not covered yet. In this note we investigate p-values based on exchangeable test statistics in order to explore reasons for the validity or failure of the Simes inequality. We provide sufficient conditions for the asymptotic validity of the Simes inequality and its possible strictness. We also show by means of an easy-to-compute counterexample that exchangeability by itself is not sufficient for the validity of the Simes inequality.
KeywordsExchangeable random variables Multivariate total positivity of order 2 Positive regression dependence Simes inequality
Mathematics Subject Classification62J15 62F03 62F05
This work was supported by the Ministry of Science and Research of the State of North Rhine-Westphalia (MIWF NRW) and the German Federal Ministry of Health (BMG).
- Heesen P (2014) Adaptive step up tests for the false discovery rate (FDR) under independence and dependence. Dissertation, University of DüsseldorfGoogle Scholar
- Sarkar TK (1969) Some lower bounds of reliability. Tech Report No 124, Dept of Operations Research and Statistics, Stanford UniversityGoogle Scholar
- Shorack GR, Wellner JA (2009) Empirical processes with applications to statistics. Classics in applied mathematics 59, SIAM edition, originally published by Wiley 1986Google Scholar