Statistical depth functions are a standard tool in nonparametric statistics to extend order-based univariate methods to the multivariate setting. Since there is no universally accepted total order for fuzzy data (even in the univariate case) and there is a lack of parametric models, a fuzzy extension of depth-based methods is very interesting. In this paper, we adapt the multivariate depths projection depth and Lr-type depth functions to the fuzzy setting, proposing different generalizations for the Lr-type depths. We prove that the proposed fuzzy depth functions have very good properties, obtaining that the fuzzy projection depth is the second example in the literature to satisfy simultaneously the notion of semilinear and of geometric depth. This implies that the fuzzy projection depth is extremely well behave, to order fuzzy sets with respect to fuzzy random variables. Furthermore, we illustrate the good empirical behavior of the proposed fuzzy depth functions with a real data example of trapezoidal fuzzy sets and the used of fuzzy depths in depth-based classification procedures. Finally, as trapezoidal fuzzy sets can be represented by elements of R4, we justify our proposals by also showing empirically the superiority of the fuzzy depths over the multivariate projection depth applied to fuzzy sets.

The recently defined concept of statistical depth function for fuzzy sets provides a theoretical framework for ordering fuzzy sets with respect to the distribution of a fuzzy random variable. One of the most used and studied statistical depth functions for multivariate data is simplicial depth, based on multivariate simplices. We introduce a notion of pseudosimplices generated by fuzzy sets and propose three generalizations of simplicial depth to fuzzy sets. Their theoretical properties are analyzed and the behavior of the proposals is illustrated through a study of both synthetic and real data.

A Vitali convergence theorem is proved for subspaces of an abstract convex combination space which admits a complete separable metric. The convergence may be in that metric or, more generally, in a quasimetric satisfying weaker properties. Versions for convergence in probability and in distribution are given. As applications, we show that some dominated convergence theorems in the literature of fuzzy random variables and random compact sets can be recovered or improved, and we derive new convergence theorems in another space of sets and in a space of probability distributions.

General properties of convergence in distribution for fuzzy random variables are studied as regards its interplay with the structure of the space of fuzzy sets. In particular, its behaviour with respect to taking tuples of fuzzy random variables, adding, multiplying by a scalar, taking the union, preserving inclusion ordering, and subsuming convergence in distribution of random sets is established.

Statistical depth functions provide a way to order the elements of a space by their centrality in a probability distribution. That has been very successful for generalizing non-parametric order-based statistical procedures from univariate to multivariate and (more recently) to functional spaces. We introduce two general definitions of statistical depth which are adapted to fuzzy data. For that purpose, two concepts of symmetric fuzzy random variables are introduced and studied. Furthermore, a generalization of Tukey's halfspace depth to the fuzzy setting is presented and proved to satisfy the above notions, through a detailed study of its properties.

Versions of several results from the theory of random variables are proved for fuzzy random variables: the Skorokhod representation theorem, the Vitali convergence theorem, the dominated convergence theorem, the continuous mapping theorem, existence of regular conditional distributions, and a few others.