Bi-log-concave Distribution Functions

Nonparametric statistics for distribution functions F or densities 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: F is said to be bi-log-concave, if both logF 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 log-concavity of f, bi-log-concavity of F allows for multimodal densities. We provide various characterisations. 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.