Expectations of random sets in Banach spaces

P. Terán (2014). Journal of Convex Analysis 21(4), to appear.


A random set is a random element of a space of sets. Since the latter are not linear (in general, you cannot subtract a set from another), it is not possible to define a notion of expectation for random sets with all the properties of the usual expectation of random variables or vectors. Thus there exist many definitions of the expectation, with various properties.

Two very interesting definitions were proposed by Aumann and Herer. Aumann's expectation is defined by putting together the expectations of all selections of the random set (i.e. if a random variable/vector is taken by selecting one point of the random set, its expectation should be an element of the expectation of the random set). Thus it is analytical in that it relies on calculating integrals in the underlying linear space.

Herer's expectation is not analytical but geometrical, as it only uses the metric structure of the space. It is defined as the locus of all points which are closer to each point x than x is, in average, to the farthest point of the random set. In other words, call R(x) the radius of a ball centered in x that covers the random set, then the Herer expectation is the intersection of all balls with center any x and radius the expected value of R(x).

The aim of the paper is to study the relationships between those two notions. Since the Herer expectation is an intersection of balls, the simplest possibility is when it is either equal to Aumann's, or the intersection of all balls covering it. The first case I had already studied, though there was a gap in the proof of the non-compact case which is corrected here.

The main types of results are:
1. Sufficient conditions on the norm for the equality between the Herer expectation and the ball hull of the Aumann expectation.
2. Sufficient conditions on the kind of sets the random set takes on as values.
3. Inclusions valid without restricting either the norm or the possible values of the random set.

The paper is a very nice amalgam of random sets, bornological differentials, and Banach space geometry. I think it makes a convincing case of how all those ingredients fit together.


Up the line:
·On the equivalence of Aumann and Herer expectations of random sets (2008).
·Intersections of balls and the ball hull mapping (2010).

Down the line:
A paper on limit theorems is in the making.


To download the paper, click on the title or here.

Centrality as a gradual notion: A new bridge between fuzzy sets and statistics

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.

On the equivalence of Aumann and Herer expectations of random sets

P. Terán (2008). Test 17, 505-514.


Trying to understand when the Herer expectation of a random set falls into the abstract definition of expectation in A law of large numbers in a metric space with a convex combination operation unexpectedly triggered a line of research on the Herer expectation itself.

This paper characterizes those separable Banach spaces where the Herer expectation is identical with the Aumann expectation, thus solving an open problem in Ilya Molchanov's book Theory of random sets as you can read here.

A few other results are presented: for instance, if the dual space is separable then the Aumann expectation is the intersection of the Herer expectations with respect to all equivalent norms. Nice!

It must be mentioned that the proof of the main theorem contains a gap and so applies only to Hausdorff approximable random sets. That does not compromise the solution of the problem as stated in Ilya's book. A different proof will be presented in forthcoming work.


Up the line:
·A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). Downloadable from Ilya's website.

Down the line:
·Intersections of balls and the ball hull mapping (2010).
·Expectations of random sets in Banach spaces (accepted 2013).

To download, click on the title or here. This preprint differs in minor details from the published version.