Bi-log-concave distribution functions

• A new shape constraint for distribution functions.
• Honest confidence bands with high precision in the tail regions.
• Honest confidence regions for moments and other functionals of a distribution.

Nonparametric statistics for distribution functions FF or densities f=F′f=F′ under qualitative shape constraints constitutes an interesting alternative to classical parametric or entirely nonparametric approaches. We contribute to this area by considering a new shape constraint: FF is said to be bi-log-concave, if both logFlogF and log(1−F)log(1−F) are concave. Many commonly considered distributions are compatible with this constraint. For instance, any c.d.f.  FF with log-concave density f=F′f=F′ is bi-log-concave. But in contrast to log-concavity of ff, bi-log-concavity of FF allows for multimodal densities. We provide various characterisations. It is shown that combining any nonparametric confidence band for FF 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 FF.