Evaluating environmental variables that vary stochastically is the principal topic for designing better environmental management and restoration schemes. Both the upper and lower estimates of these variables, such as water quality indices and flood and drought water levels, are important and should be consistently evaluated within a unified mathematical framework. We propose a novel pair of Orlicz regrets to consistently bound the statistics of random variables both from below and above. Here, consistency indicates that the upper and lower bounds are evaluated with common coefficients and parameter values being different from some of the risk measures proposed thus far. Orlicz regrets can flexibly evaluate the statistics of random variables based on their tail behavior. The explicit linkage between Orlicz regrets and divergence risk measures was exploited to better comprehend them. We obtain sufficient conditions to pose the Orlicz regrets as well as divergence risk measures, and further provide gradient descent-type numerical algorithms to compute them. Finally, we apply the proposed mathematical framework to the statistical evaluation of 31-year water quality data as key environmental indicators in a Japanese river environment.

翻译：评估随机变化的环境变量是设计更优环境管理与修复方案的核心课题。水质量指数、洪水与干旱水位等变量的上下估计量均至关重要，且需在统一数学框架内进行一致性评估。我们提出一对新颖的奥尔利奇遗憾，用于从下方和上方对随机变量的统计量进行一致有界约束。此处，一致性指上下界通过共享系数与参数值进行评估，这与迄今提出的部分风险度量方法有所不同。奥尔利奇遗憾可基于随机变量的尾部行为灵活评估其统计量。我们揭示了奥尔利奇遗憾与散度风险度量之间的显式关联，以更深入理解两者。我们获得了构造奥尔利奇遗憾及散度风险度量的充分条件，并进一步提出了梯度下降型数值算法用于计算。最后，我们将所提出的数学框架应用于日本某河流环境31年水质数据（作为关键环境指标）的统计评估。