P. Terán (2011). International Journal of Approximate Reasoning 52, 1243-1256.
According to one point of view, fuzzy set theoretical notions are problematic unless they can be justified as / explained from / reduced to ordinary statistics and probability. I can't say that this makes much sense to me.
In this paper the opposite route is taken, which is fun. It subverts that view by writing a similar paper in which statistical/probabilistic notions are reduced to fuzzy ones. The point is: So what?
A fuzzy set of central points of a probability distribution with respect to a family of fuzzy reference events is defined. Its fuzzy set theoretical interpretation is very natural: the membership degree of x equals the truth value of the proposition "Every reference event containing x is probable".
Also natural location estimators are the points whose membership in that fuzzy set is maximal. The paper presents many examples of known notions from statistics and probability arising as maximally central estimators (of a distribution or, more generally, of a family of distributions). The prototype of a maximally central estimator is the mode (taking the singletons as reference events), and MCEs can thus be seen as generalized modes.
From the paper's abstract: "This framework has a natural interpretation in terms of fuzzy logic and unifies many known notions from statistics, including the mean, median and mode, interquantile intervals, the Lorenz curve, the halfspace median, the zonoid and lift zonoid, the coverage function and several expectations and medians of random sets, and the Choquet integral against an infinitely alternating or infinitely monotone capacity."
Up the line:
This starts a new line.
Down the line:
·Connections between statistical depth functions and fuzzy sets (2010).
A long paper on statistical consistency has been submitted.
To download, click on the title or here.