Bi-$s^{\ast }$-concave distributions

Abstract

We introduce a new shape-constrained class of distribution functions on R, the bi-$s^{\ast }$-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-$s^{\ast }$-concave distribution function $F$ and that every bi-$s^{\ast }$-concave distribution function satisfies $\gamma (F)\le 1/(1+s)$ where finiteness of $$\gamma (F)\equiv \underset{x}{\sup }F(x)(1-F(x))\frac{|f^{\prime }(x)|}{f^2(x)},$$ the Csörgő - Révész constant of F, plays an important role in the theory of quantile processes on $R$.

30 pages, 11 figures