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arXiv · 2006.03989

Bi-$s^*$-Concave Distributions

Nilanjana Laha·Zhen Miao·Jon A. Wellner
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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.

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Nilanjana Laha, Zhen Miao, Jon A. Wellner. 2020-06-06. Bi-$s^*$-Concave Distributions. https://arxiv.org/abs/2006.03989

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