We introduce new shape-constrained classes of distribution functions on $\mathbb{R}$, the bi-$s^{\ast }$-concave classes. In parallel to results of Dümbgen et al. (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 confidence 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.

我们引入了新的形状约束分布函数类$\mathbb{R}$，双${秒}^{＊}$-凹类。与 Dümbgen 等人的结果平行。(2017) 对于他们所谓的双对数凹分布函数类，我们表明每个$秒$-凹密度$F$有一个双${秒}^{＊}$-凹分布函数$F$为了${秒}^{＊}\le 秒∕(秒+\mathrm{1个})$.

置信带建立在现有的非参数置信带之上，但考虑了双向的形状约束${秒}^{＊}$-凹性，也被考虑在内。新频段扩展了 Dümbgen 等人开发的频段。(2017) 用于双对数凹性的约束。我们还建立了双${秒}^{＊}$-Csörgő–Révész 常数的凹性和有限性$F$这在分位数过程理论中起着重要作用。