Fuzzy partitions of the sample space and fuzzy parameter hypotheses

In this paper, a generalization of parameter hypotheses tests used in statistical inference is presented. This leads to fuzzy hypotheses defined as imprecise predictions about the unknown parameter of a family of distribution functions in question, rather than exactly and sharply given assertions about its location in the parameter space. The notion of fuzzy partitions is discussed in the first part of this paper and details regarding their structure in relation to possibility theoretical interpretations are investigated subsequently. In the second part the authors introduce the notions of fuzzy tests and disclose the impacts of an ill defined counter hypothesis (i.e., hypotheses, partly supporting the same subset of the parameter space) on the reliability of such a fuzzy test. An example demonstrates the consistency of the notion of fuzzy tests with the classical crisp case in statistical inference