Heterogeneous compound $p$-values and closure principle

The closure principle applied to FDR control have been recently studied and has been shown to improve former FDR control methods, as the BY procedure or the BH procedure. In this paper, we show that the closure principle can be applied to heterogeneous compounds $p$-values, and therefore improve former FDR control methods designed for homogeneous $p$-values, such as the closed BY procedure. Hence considering heterogeneity within the closure principle derive less conservative methods. These heterogeneous compounds $p$-values allows to retrieve former FDR control methods and to extend their validity to more general settings.