Optimal Hypothesis Testing in Case of Regulatory Thresholds

Abstract

In this study hypothesis testing is treated, when neither the probability density function (pdf) of the test statistic under the null hypothesis nor the pdf of the test statistic under the alternative hypothesis are known. First, the classical procedure in case of random variability is reviewed. Then, the testing procedure is extended to the case when the uncertainty of the measurements comprises both random and systematic errors. Both types of uncertainty are treated in a comprehensive way using fuzzy-random variables (FRVs) which represent a combination of probability and fuzzy theory. The classical case of random errors (absence of systematic errors) is a special case of FRVs. The underlying theory of the procedure is outlined in particular. The approach allows the consideration of fuzzy regions of acceptance and rejection. The final (optimal) test decision is based on the utility theory which selects the test decision with the largest expected utility as the most beneficial one. An example illustrates the theoretical concept.