Abstract
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.
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