The Wabl/Ldev/Rdev Median of a Random Fuzzy Number and Statistical Properties

Study of the choice of the weighting measure φ on the φ-wabl/ldev/rdev median

“…The recently introduced ϕ-wabl/ldev/rdev median for random fuzzy numbers (see [16]) has been recalled, as well as the distance this notion is based on. Two properties of such metric have been proved in this manuscript.…”

“…The ϕ-wabl/ldev/rdev median of a random fuzzy number fulfills most of the basic properties of the median of a random variable, like the equivariance under 'linear' transformations, the symmetry about a real number when the random fuzzy number is symmetric and the strong consistency of the sample version under some sufficient conditions (for more details, see [16]). With respect to the robustness of this proposal, the finite sample breakdown point of the sample ϕ-wabl/ldev/rdev median was calculated.…”

“…In order to recall how this generalization has been done in the literature, the L 1 distance based on the ϕ-wabl/ldev/rdev representation and used in such concept will be specified now. [16] Given an absolutely continuous probability measure ϕ on the measurable space 1] ) with positive mass function on (0, 1) and a parameter θ ∈ (0, 1], the wabl/ldev/rdev-based…”

“…Theorem 3.1 [16] Given a probability space (Ω, A, P ), an absolutely continuous probability measure ϕ on the measurable space ([0, 1], B [0,1] ) with positive mass function on (0, 1) and an associated RFN X , for any α ∈ [0, 1], the fuzzy number…”

“…In detail, [13] and [14] use L 1 metrics that make use of representations of fuzzy numbers for which necessary and sufficient conditions to characterize them are known, so the median can be defined as the fuzzy number minimizing the mean distance to all the values the RFN takes. A third alternative, the one considered in this paper, was introduced in [16] as a generalization of the Hausdorff-type median for random intervals (see [12]) following the same scheme.…”