Subuniformity of harmonic mean p‐values

Abstract We obtain several inequalities on the generalized means of dependent ‐values. In particular, the weighted harmonic mean of ‐values is strictly subuniform under several dependence assumptions of ‐values, including independence, negative upper orthant dependence, the class of extremal mixture copulas, and some Clayton copulas. Subuniformity of the harmonic mean of ‐values has an important implication in multiple hypothesis testing: It is statistically invalid (anticonservative) to merge ‐values using the harmonic mean unless a proper threshold or multiplier adjustment is used, and this applies across all significance levels. The required multiplier adjustment on the harmonic mean ‐value grows sublinearly to infinity as the number of ‐values increases, and hence there does not exist a constant multiplier that works for any number of ‐values, even under independence.