Bi-*-Concave Distributions.

We introduce new shape-constrained classes of distribution functions on , the bi-*-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 -concave density has a bi-*-concave distribution function for * ≤ /( + 1). Confidence bands building on existing nonparametric confidence bands, but accounting for the shape constraint of bi-*-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-*-concavity and finiteness of the Csörgő - Révész constant of which plays an important role in the theory of quantile processes.