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

TL;DR

This paper introduces Orlicz regrets, a new mathematical framework for consistently bounding the statistics of environmental variables, demonstrated through a 31-year water quality data analysis in Japan.

Contribution

It proposes a novel pair of Orlicz regrets for unified, consistent bounds on stochastic environmental variables, linking them to divergence risk measures and providing computational algorithms.

Findings

Successfully applied to 31-year water quality data in Japan

Provided explicit conditions for bounds and risk measures

Developed gradient descent algorithms for computation

Abstract

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.