Distribution of the Harmonic Mean of P-values with Application to Multiple Testing

Abstract In this paper we analyze the null distribution of the harmonic mean of i.i.d. P -values, each distributed as U [0, 1]. We derive the exact distribution numerically for small n using the convolution method and asymptotic distribution for large n . We show that the critical constants computed using the exact distribution and the asymptotic distribution match fairly well. We construct a procedure based on the harmonic means of P -value for testing multiple hypotheses using the closure principle (Marcus et al., Biometrika , 63 , 655–660, 1976). We show via simulation that our proposed HMP procedure is generally more powerful than several other P -value based procedures including the Holm ( Scandinavian Journal of Statistics , 6, 65–70, 1979), Hochberg ( Biometrika , 75, 800–802, 1988), Hommel ( Biometrika , 75, 383–386, 1988) and the Hochberg-Hommel hybrid procedure (Gou et al., Biometrika , 101, 899–911, 2014) under independence as well as under dependence (both positive and negative). It can be implemented using a simple code.