Abstract:
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We introduce a new shape-constrained class of univariate distribution functions, the bi-s*-concave class. In parallel to results of D\"umbgen, Kolesnyk, and Wilke [Journal of Statistical Planning and Inference, 184, 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. We establish that the Cs\"org\H{o} - R\'ev\'esz constant of F is finite for every bi-s*-concave distribution function where the Cs\"org\H{o} - R\'ev\'esz constant plays an important role in the theory of quantile processes on the real line. We also construct s*-concave confidence bands for F refining some non-parametric confidence bands.
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