Fuzzy hypothesis testing for a population proportion based on set-valued information

Abstract

In this paper we propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one-tailed hypergeometric test with fuzzy (non-)complementary hypotheses, and incorporate the conventional hypergeometric test with crisp hypotheses into its framework. In particular, we formulate hypotheses as ontic or epistemic (fuzzy) sets, since we assume that presumptions regarding the true population proportion can be expressed in the conjunctive or disjunctive reading. Further, we suggest modeling of membership functions in terms of cost or frequency due to considerations of user's priorities or incomplete knowledge in relation to hypotheses formulation. Considering a hypothesis as union of its crisp and fuzzy areas, we illustrate via real-life examples that in contrast to classical test theory, fuzzy hypothesis testing provides an additional partial and gradual consideration of the indifference zone for both complementary and non-complementary hypotheses. The test concept is introduced for both test methods, a test of significance and an alternative test, and leads to a crisp test decision considering further cost aspects like the sample size. The generalized error criteria are derived and interpreted in relation to both hypotheses. Additionally, a corresponding sensitivity analysis is presented for one of the most promising test types.