The likelihood ratio test for goodness of fit with fuzzy experimental observations

Experiments are considered in which the person responsible for observation cannot crisply perceive the outcomes, but where each observable event may be identified with a fuzzy subset of the sample space. (More precisely, one may classify the experimental observation according to fuzzy information as intended by Zadeh and Tanaka et al.) In such a situation, the likelihood ratio test for goodness of fit to a completely specified hypothetical distribution regarding the ‘exact experiment’ on the basis of fuzzy information can immediately be derived. On the other hand, if the hypothetical distribution involves unknown parameters the extension of the likelihood ratio test usually becomes unmanageable, because of the unoperativeness of the trivial generalization of the maximum likelihood principle to fuzzy observations. Nevertheless, this last generalization is suitably approximated by means of the minimum inaccuracy principle of point estimation (introduced in previous papers as an operative extension of the maximum likelihood one, on the basis of the well-known inaccuracy measure defined by Kerridge) whose use for the likelihood ratio test for goodness of fit with fuzzy data provides us with a very manageable procedure.