On statistical inference for fuzzy data with applications to descriptive statistics

The object of this paper is to discuss how fuzziness of data is propagated when statistical inference for samples of non-precise data is carried out. First, "fuzzy data" and "fuzzy samples" are defined. Subsequently, the method of propagation of fuzziness as used in previous work for specific stochastic models is formulated in rather general terms. This method may be applied to any statistical method which leads to a result that may be expressed as a function f(x1,..., xn) of the data x1,..., xn. It is proved that this method is equal to determining the images of a family of compact subsets of the sample space under the function f(·). This equivalence is utilized in order to perform propagation of fuzziness in practice. Finally, this approach is applied to some concepts of descriptive statistics such as fuzzy sample mean, fuzzy sample variance and to the empirical distribution function of a fuzzy sample.