  Several classical methods exist for controlling the false discovery
exceedance (FDX) for large scale multiple testing problems, among them the
Lehmann-Romano procedure ([LR] below) and the Guo-Romano procedure ([GR]
below). While these two procedures are the most prominent, they were originally
designed for homogeneous test statistics, that is, when the null distribution
functions of the $p$-values $F_i$, $1\leq i\leq m$, are all equal. In many
applications, however, the data are heterogeneous which leads to heterogeneous
null distribution functions. Ignoring this heterogeneity usually induces a
conservativeness for the aforementioned procedures. In this paper, we develop
three new procedures that incorporate the $F_i$'s, while ensuring the FDX
control. The heterogeneous version of [LR], denoted [HLR], is based on the
arithmetic average of the $F_i$'s, while the heterogeneous version of [GR],
denoted [HGR], is based on the geometric average of the $F_i$'s. We also
introduce a procedure [PB], that is based on the Poisson-binomial distribution
and that uniformly improves [HLR] and [HGR], at the price of a higher
computational complexity. Perhaps surprisingly, this shows that, contrary to
the known theory of false discovery rate (FDR) control under heterogeneity, the
way to incorporate the $F_i$'s can be particularly simple in the case of FDX
control, and does not require any further correction term. The performances of
the new proposed procedures are illustrated by real and simulated data in two
important heterogeneous settings: first, when the test statistics are
continuous but the $p$-values are weighted by some known independent weight
vector, e.g., coming from co-data sets; second, when the test statistics are
discretely distributed, as is the case for data representing frequencies or
counts.
