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议题。这些变量的上下界估计（如水质指数、洪水与干旱水位）均至关重要，且应在统一的数学框架内进行一致性评价。我们提出一对新颖的Orlicz遗憾函数，用以从上下两个方向一致性界定随机变量的统计量。此处，“一致性”指上下界采用共同系数与参数值进行评价，这不同于现有部分风险测度方法。Orlicz遗憾函数可根据随机变量的尾部行为灵活评估其统计量。我们揭示了Orlicz遗憾函数与散度风险测度之间的显式关联，以增进对其的理解。我们给出了Orlicz遗憾函数及散度风险测度成立的条件，并提出了基于梯度下降的数值算法进行求解。最后，我们将所提出的数学框架应用于日本某河流环境31年水质数据这一关键环境指标的统计评估。