Fuzzy probabilities and their applications to statistical inference

摘要

We discuss the issue of probability of a fuzzy event. The value of this probability is a fuzzy set rather than a number between zero and one. We compare this concept with the non fuzzy probability of a fuzzy event and we investigate some possible shortcomings of the latter. Explicit formulae for the fuzzy probability are given in both the discrete and continuous cases. In particular, in the discrete uniform case we relate the concepts of fuzzy probability and fuzzy cardinality. Our concept of fuzzy probability is applied to statistical inference and in particular to testing of hypotheses of the form " theta is F", where F is a fuzzy set. The level of significance and the power function of testing procedures are evaluated as fuzzy probabilities.