A New Bridge Between Fuzzy Sets and Statistics

Our basic question is as follows: given a, maybe multivariate, probability distribution, how to formalize the idea that some points are more central than others? We present an account of the notion of centrality which is based on fuzzy events and is valid for single distributions and for families of distributions (including imprecise probability models). This unifying framework is natural and subsumes many known concepts like the following (not an exhaustive list): (a) univariate location estimators like the mean, the median and the mode, (b) the interquartile interval and the Lorenz curve of a random variable, (c) several generalized medians, trimmed regions and statistical depth functions from multivariate analysis, (d) most known location estimators for random sets, (e) the probability mass function of a discrete random variable and the coverage function of a random closed set, (f) the Choquet integral with respect to an infinitely alternating or infinitely monotone capacity. Keywords— Centrality, Fuzzy event, Multivariate analysis, Random set, Statistical depth.