Bi-$s^*$-concave distributions

TL;DR

This paper introduces the bi-$s^*$-concave class of distributions, exploring their properties and connections to s-concave densities, with implications for quantile process theory.

Contribution

It defines the bi-$s^*$-concave class, establishes its relation to s-concave densities, and analyzes the Cs"org ext{"o}-Révész constant within this context.

Findings

Every s-concave density has a bi-$s^*$-concave distribution function.

Bi-$s^*$-concave distribution functions satisfy $\gamma(F) \le 1/(1+s)$.

The Cs"org ext{"o}-Révész constant $\gamma(F)$ is crucial in quantile process theory.

Abstract

We introduce a new shape-constrained class of distribution functions on R, the bi-$s^*$-concave class. In parallel to results of D\"umbgen, 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 $F$ and that every bi-$s^*$-concave distribution function satisfies $\gamma (F) \le 1/(1+s)$ where finiteness of $$ \gamma (F) \equiv \sup_{x} F(x) (1-F(x)) \frac{| f' (x)|}{f^2 (x)}, $$ the Cs\"org\H{o} - R\'ev\'esz constant of F, plays an important role in the theory of quantile processes on $R$.