The minimum inaccuracy estimates in \(\chi ^ 2\) tests for goodness of fit with fuzzy observations
Experiments, where fuzzy information (fi) \(x_ k\) is obtained from X, are characterized. The information can be classified into classes and a fuzzy partition is determined. Under a hypothetical probability measure P the probability of the fi is calculable. An unknown probability vector is associated with the classes and fixes the behavior of the true probability of the fi system. Once a sample is selected the goodness of fit problem arises. The classical results of \(\chi^ 2\)-tests are reanalyzed in a fuzzy-set environment. The extension is straightforward and Pearson's statistic is used with a similar interpretation. An example fixes the similarity between the usual meaning of large values of the statistic in the classical frame and in the fuzzy one. When some parameters are unknown the corresponding results are stated by using a likelihood function. This agrees with the structure of a measure of inaccuracy (mi) of order \(\beta =2\) in information theory. The concept of minimum inaccuracy is used for deriving an estimator. It looks as an mi of order \(\beta =1\). Under some mild regularity conditions an estimator attaining the lower bound of the variance is the unique optimum estimate if some assumptions are fulfilled. Then estimators for some distributions are given. The mi-estimate for the \(\chi^ 2\)-statistic of goodness of fit is derived. An example illustrates its behavior.
scientific article; zbMATH DE number 4088702