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.

中文翻译：

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基于集合值信息的总体比例模糊假设检验

摘要 在本文中，我们提出了具有清晰数据的比例的模糊假设检验作为具有模糊（非）互补假设的精确广义单尾超几何检验，并将具有清晰假设的传统超几何检验纳入其框架。特别是，我们将假设表述为本体或认知（模糊）集合，因为我们假设关于真实人口比例的假设可以在联合或分离阅读中表达。此外，由于考虑到用户的优先级或与假设制定相关的不完整知识，我们建议在成本或频率方面对成员函数进行建模。将假设视为其清晰和模糊区域的结合，我们通过现实生活中的例子来说明，与经典测试理论相比，模糊假设检验为互补和非互补假设提供了对无差异区的额外部分和渐进的考虑。测试概念引入了两种测试方法、显着性测试和替代测试，并导致考虑到样本大小等其他成本方面的清晰测试决策。广义误差标准是根据这两个假设导出和解释的。此外，还针对最有希望的测试类型之一进行了相应的敏感性分析。并考虑到样本大小等其他成本因素，从而做出清晰的测试决策。广义误差标准是根据这两个假设导出和解释的。此外，还针对最有希望的测试类型之一进行了相应的敏感性分析。并考虑到样本大小等其他成本因素，从而做出清晰的测试决策。广义误差标准是根据这两个假设导出和解释的。此外，还针对最有希望的测试类型之一进行了相应的敏感性分析。