Biom J. 2021 Apr 9. doi: 10.1002/bimj.202000207. Online ahead of print.
ABSTRACT
In a paper published in 1939 in The Annals of Mathematical Statistics, Wald and Wolfowitz discussed the possible validity of a probability inequality between one- and two-sided coverage probabilities for the empirical distribution function. Twenty-eight years later, Vandewiele and Noé proved this inequality for Kolmogorov-Smirnov type goodness of fit tests. We refer to this type of inequality as one-two inequality. In this paper, we generalize their result for one- and two-sided union-intersection tests based on positively associated random variables and processes. Thereby, we give a brief review of different notions of positive association and corresponding results. Moreover, we introduce the notion of one-two dependence and discuss relationships with other dependence concepts. While positive association implies one-two dependence, the reverse implication fails. Last but not least, the Bonferroni inequality and the one-two inequality yield lower and upper bounds for two-sided acceptance/rejection probabilities which differ only slightly for significance levels not too large. We discuss several examples where the one-two inequality applies. Finally, we briefly discuss the possible impact of the validity of a one-two inequality on directional error control in multiple testing.
PMID:33837570 | DOI:10.1002/bimj.202000207
