title: Implications of Stein's Paradox for Environmental Standard Compliance Assessment.
abstract: The implications of Stein's paradox stirred considerable debate in statistical circles when the concept was first introduced in the 1950s. The paradox arises when we are interested in estimating the means of several variables simultaneously. In this situation, the best estimator for an individual mean, the sample average, is no longer the best. Rather, a shrinkage estimator, which shrinks individual sample averages toward the overall average is shown to have improved overall accuracy. Although controversial at the time, the concept of shrinking toward overall average is now widely accepted as a good practice for improving statistical stability and reducing error, not only in simple estimation problems, but also in complicated modeling problems. However, the utility of Stein's insights are not widely recognized in the environmental management community, where mean pollutant concentrations of multiple waters are routinely estimated for management decision-making. In this essay, we introduce Stein's paradox and its modern generalization, the Bayesian hierarchical model, in the context of environmental standard compliance assessment. Using simulated data and nutrient monitoring data from wadeable streams around the Great Lakes, we show that a Bayesian hierarchical model can improve overall estimation accuracy, thereby improving our confidence in the assessment results, especially for standard compliance assessment of waters with small sample sizes.
authors: †Department of Environmental Sciences, The University of Toledo, Toledo, Ohio 43606, United States.
authors: Song S
authors: SS
authors: Qian
authors: ‡Great Lakes Environmental Research Laboratory, National Oceanic and Atmospheric Administration, Ann Arbor, Michigan 48108, United States.
authors: Craig A
authors: CA
authors: Stow
authors: §School of Natural Resources and Environment, University of Michigan, Ann Arbor, Michigan 48108, United States.
authors: YoonKyung
authors: Y
authors: Cha
mesh_terms: D001499:Bayes Theorem; D004777:Environment; D033161:Government Regulation; D008962:Models, Theoretical; D014876:Water Pollution / Q000331:legislation & jurisprudence / Q000706:statistics & numerical data*; D060753:Water Quality