arXiv subjects

Pedro Terán

Publications and source records attributed to Pedro Terán.

6 recordsLinked to original sources

arXiv2026

On the continuity of the Tukey depth function for fuzzy data

The practical use of statistical depth for fuzzy data requires regularity, inferential stability, and computational feasibility. This paper studies these aspects for the first time for the fuzzy depth, in particular for the fuzzy Tukey depth. In particular, we investigate the continuity of this depth with respect to each of its arguments. As a function of the elements of the underlying space, we prove its upper semicontinuity under the main metrics used in fuzzy spaces. As a function of the distribution with respect to which the depth is computed, we establish the almost sure uniform consistency of its empirical version for continuous fuzzy random variables. These results ensure closed depth regions and support the use of sample depth values to approximate their population counterparts. We also derive consistency of empirical deepest-point estimators and study a finite-grid approximation of the depth, supported by theoretical and simulation results.

math.ST↗

arXiv2025

On independence and large deviations for sublinear expectations

We prove by counterexample that a large deviation principle established by Chen and Feng [{\em Comm. Statist. Theory Methods} {\bf 45} (2016), 400--412] in the framework of sublinear expectations is incorrect. That implies that the rate function cannot, in general, be obtained by computing the Fenchel transform of the cumulant generating function, as is the case for ordinary probabilities. We derive a corrected version of that result and show that the original presentation holds under a stronger independence assumption.

math.PR↗

arXiv2023

Statistical depth and support medians for fuzzy data

Statistical depth functions order the elements of a space with respect to their centrality in a probability distribution or dataset. Since many depth functions are maximized in the real line by the median, they provide a natural approach to defining median-like location estimators for more general types of data (in our case, fuzzy data). We analyze the relationships between depth-based medians, medians based on the support function, and some notions of a median for fuzzy data in the literature. We take advantage of specific depth functions for fuzzy data defined in our former papers: adaptations of Tukey depth, simplicial depth, $L^1$-depth and projection depth.

math.ST↗

arXiv2023

Projection depth and $L^r$-type depths for fuzzy random variables

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 projection depth and $L^{r}$-type depth to the fuzzy setting, studying their properties and illustrating their behaviour with a real data example.

math.ST↗

arXiv2022

Properties of statistical depth with respect to compact convex random sets. The Tukey depth

We study a statistical data depth with respect to compact convex random sets which is consistent with the multivariate Tukey depth and the Tukey depth for fuzzy sets. In doing so, we provide a series of properties for statistical data depth with respect to compact convex random sets. These properties are an adaptation of properties that constitute the axiomatic notions of multivariate, functional and fuzzy depth functions and other well-known properties of depth.

math.ST↗

arXiv2022

Simplicial depths for fuzzy random variables

The recently defined concept of a 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 function for multivariate data is simplicial depth, based on multivariate simplices. We introduce a notion of pseudosimplices generated by fuzzy sets and propose three plausible generalizations of simplicial depth to fuzzy sets. Their theoretical properties are analyzed and the behavior of the proposals illustrated through a study of both synthetic and real data.

stat.ME↗