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.