Nonparametric statistics for distribution functions F or densities f=F' under
qualitative shape constraints provides an interesting alternative to classical
parametric or entirely nonparametric approaches. We contribute to this area by
considering a new shape constraint: F is said to be bi-log-concave, if both
log(F) and log(1 - F) are concave. Many commonly considered distributions are
compatible with this constraint. For instance, any c.d.f. F with log-concave
density f = F' is bi-log-concave. But in contrast to the latter constraint,
bi-log-concavity allows for multimodal densities. We provide various
characterizations. It is shown that combining any nonparametric confidence band
for F with the new shape-constraint leads to substantial improvements,
particularly in the tails. To pinpoint this, we show that these confidence
bands imply non-trivial confidence bounds for arbitrary moments and the moment
generating function of F
