Bi-$s^{\ast }$-Concave Distributions

Abstract

We introduce new shape-constrained classes of distribution functions on R, the bi-$s^{\ast }$-concave classes. 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-$s^{\ast }$-concave distribution function $F$ for $s^{\ast }\le s/(s+1)$. Confidence bands building on existing nonparametric bands, but accounting for the shape constraint of bi-$s^{\ast }$-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-$s^{\ast }$-concavity and finiteness of the Csörgő - Révész constant of $F$ which plays an important role in the theory of quantile processes.

68 pages, 24 figures; replaces and extends arXiv:2006.03989 by Laha, Miao, and Wellner