Bi-$s^*$-Concave Distributions

TL;DR

This paper introduces bi-$s^*$-concave distribution classes, explores their properties, and develops confidence bands that incorporate shape constraints, extending previous work on bi-log-concavity.

Contribution

It defines new bi-$s^*$-concave classes, establishes their relation to $s$-concave densities, and extends confidence band methods to these shape-constrained distributions.

Findings

Every $s$-concave density has a bi-$s^*$-concave distribution function for $s^* \\leq s/(s+1)$.

Developed confidence bands accounting for bi-$s^*$-concavity.

Connected bi-$s^*$-concavity with the finiteness of the Cs"org ext{"o}-Révész constant.

Abstract

We introduce new shape-constrained classes of distribution functions on R, the bi-$s^*$-concave classes. 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 $F$ for $s^*\leq s/(s+1)$. Confidence bands building on existing nonparametric bands, but accounting for the shape constraint of bi-$s^*$-concavity, are also considered. The new bands extend those developed by D\"umbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi-$s^*$-concavity and finiteness of the Cs\"org\H{o} - R\'ev\'esz constant of $F$ which plays an important role in the theory of quantile processes.