How to relate fuzzy and OWA estimates

In many practical situations, we have several estimates x
₁
, ..., x
_n
of the same quantity x, i.e., estimates for which x
₁
≈ x, x
₂
≈ x, ..., and x
_n
≈ x. It is desirable to combine (fuse) these estimates into a single estimate for x. From the fuzzy viewpoint, a natural way to combine these estimates is: (1) to describe, for each x and for each i, the degree μ≈(x
_i
-x) to which x is close to x
_i
, (2) to use a t-norm (“and”-operation) to combine these degrees into a degree to which x is consistent with all n estimates, and then (3) find the estimate x for which this degree is the largest. Alternatively, we can use computationally simpler OWA (Ordered Weighted Average) to combine the estimates x
_i
. To get better fusion, we must appropriately select the membership function μ≈(x), the t-norm (in the fuzzy case) and the weights (in the OWA case). Since both approaches - when applied properly - lead to reasonable data fusion, it is desirable to be able to relate the corresponding selections. For example, once we have found the appropriate μ≈(x) and t-norm, we should be able to deduce the appropriate weights - and vice versa. In this paper, we describe such a relation. It is worth mentioning that while from the application viewpoint, both fuzzy and OWA estimates are not statistical, our mathematical justification of the relation between them uses results that have been previously applied to mathematical statistics.