Environmental data analysis based on the gamma distribution: compliance assessment using tolerance limits and exceedance fractions

Abstract

The gamma distribution has found extensive applications in the modeling and analysis of environmental data. Furthermore, for groundwater monitoring and environmental monitoring, an upper tolerance limit (i.e., an upper confidence limit for a percentile) can be used for checking compliance and is recommended in some of the standards published by the U.S. EPA. In addition, estimation of the proportion of samples where the measurements exceed a threshold (proportion of non‐compliance, for example), referred to as an exceedance fraction, is of obvious interest. Thus the computation of an accurate upper tolerance limit, and an accurate upper confidence limit for the exceedance fraction, are important. The problems are somewhat challenging since environmental samples are typically not large, and large sample methods cannot be used. In the present investigation, we have succeeded in obtaining accurate solutions to the above problems using a small sample modification of the likelihood based method. The results are also applied for analyzing a data set on vinyl chloride concentrations in ground water monitoring wells.