Abstract
Abstract We introduce new shape-constrained classes of distribution functions on R , the bi- s ∗ -concave classes. In parallel to results of Dumbgen 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 Dumbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi- s ∗ -concavity and finiteness of the Csorgő - Revesz constant of F which plays an important role in the theory of quantile processes.
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