We investigate the cumulative Tsallis entropy, an information measure recently introduced as a cumulative version of the classical Tsallis differential entropy, which is itself a generalization of the Boltzmann-Gibbs statistics. This functional is here considered as a perturbation of the expected mean residual life via some power weight function. This point of view leads to the introduction of the dual cumulative Tsallis entropy and of two families of coherent risk measures generalizing those built on mean residual life. We characterize the finiteness of the cumulative Tsallis entropy in terms of ${\mathcal L}_p$-spaces and show how they determine the underlying distribution. The range of the functional is exactly described under various constraints, with optimal bounds improving on all those previously available in the literature. Whereas the maximization of the Tsallis differential entropy gives rise to the classical $q-$Gaussian distribution which is a generalization of the Gaussian having a finite range or heavy tails, the maximization of the cumulative Tsallis entropy leads to an analogous perturbation of the Logistic distribution.

翻译：我们研究累积Tsallis熵，这是一种最近被引入的信息度量，作为经典Tsallis微分熵的累积版本，而Tsallis微分熵本身是Boltzmann-Gibbs统计量的推广。本函数在此被视为通过某种幂权重函数对期望平均剩余寿命的扰动。这一观点引出了对偶累积Tsallis熵以及两类基于平均剩余寿命构建的相干风险测度族的引入。我们利用${\mathcal L}_p$空间刻画了累积Tsallis熵的有界性，并展示了它们如何决定潜在分布。在不同约束条件下精确描述了该泛函的取值范围，其最优界改进了文献中此前已有的所有结果。尽管Tsallis微分熵的最大化产生了经典的$q-$高斯分布（一种具有有限支撑或重尾的高斯分布的推广），而累积Tsallis熵的最大化则导出了Logistic分布的类似扰动形式。