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Evaluating environmental variables that vary stochastically is the principaltopic for designing better environmental management and restoration schemes.Both the upper and lower estimates of these variables, such as water qualityindices and flood and drought water levels, are important and should beconsistently evaluated within a unified mathematical framework. We propose anovel pair of Orlicz regrets to consistently bound the statistics of randomvariables both from below and above. Here, consistency indicates that the upperand lower bounds are evaluated with common coefficients and parameter valuesbeing different from some of the risk measures proposed thus far. Orliczregrets can flexibly evaluate the statistics of random variables based on theirtail behavior. The explicit linkage between Orlicz regrets and divergence riskmeasures was exploited to better comprehend them. We obtain sufficientconditions to pose the Orlicz regrets as well as divergence risk measures, andfurther provide gradient descent-type numerical algorithms to compute them.Finally, we apply the proposed mathematical framework to the statisticalevaluation of 31-year water quality data as key environmental indicators in aJapanese river environment.

Orlicz regrets to consistently bound statistics of random variables with an application to environmental indicators