Bi- s *-Concave Distributions.

We introduce new shape-constrained classes of distribution functions on R , the bi- s *-concave classes. In parallel to results of Dümbgen et al. (2017) for what they called the class of bi-log-concave distribution functions, we show that every s -concave density f has a bi- s *-concave distribution function F for s * ≤ s /( s + 1). Confidence bands building on existing nonparametric confidence bands, but accounting for the shape constraint of bi- s *-concavity, are also considered. The new bands extend those developed by Dümbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi- s *-concavity and finiteness of the Csörgő - Révész constant of F which plays an important role in the theory of quantile processes.