Bi-s^*-Concave Distributions
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We introduce a new shape-constrained class of distribution functions on R, the bi-s^*-concave class. In parallel to results of Dümbgen, Kolesnyk, and Wilke (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) and that every bi-s^*-concave distribution function satisfies γ(F)≤ 1-s^* where finiteness of γ(F)≡_x∈ J(F) F(x)(1-F(x))|f^'(x)|/f^2(x), the Csörgő - Révész constant of F, plays an important role in the theory of quantile processes on R. 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, Kolesnyk, and Wilke (2017) for the constraint of bi-log-concavity.
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