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On Poincaré cone property

Publicated to:ANNALS OF STATISTICS. 42 (1): 255-284 - 2014-02-01 42(1), DOI: 10.1214/13-AOS1188

Authors: Cholaquidis, Alejandro; Cuevas, Antonio; Fraiman, Ricardo

Affiliations

Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain - Author
Univ Republica, Ctr Matemat, Montevideo 11400, Uruguay - Author
Universidad Autónoma de Madrid - Author
Universidad de la República - Author

Abstract

A domain S ⊂ ℝd is said to fulfill the Poincaré cone property if any point in the boundary of S is the vertex of a (finite) cone which does not otherwise intersects the closure S̄. For more than a century, this condition has played a relevant role in the theory of partial differential equations, as a shape assumption aimed to ensure the existence of a solution for the classical Dirichlet problem on S. In a completely different setting, this paper is devoted to analyze some statistical applications of the Poincaré cone property (when defined in a slightly stronger version). First, we show that this condition can be seen as a sort of generalized convexity: while it is considerably less restrictive than convexity, it still retains some "convex flavour." In particular, when imposed to a probability support S, this property allows the estimation of S from a random sample of points, using the "hull principle" much in the same way as a convex support is estimated using the convex hull of the sample points. The statistical properties of such hull estimator (consistency, convergence rates, boundary estimation) are considered in detail. Second, it is shown that the class of sets fulfilling the Poincaré property is a P-Glivenko-Cantelli class for any absolutely continuous distribution P on ℝd. This has some independent interest in the theory of empirical processes, since it extends the classical analogous result, established for convex sets, to a much larger class. Third, an algorithm to approximate the cone-convex hull of a finite sample of points is proposed and some practical illustrations are given.

Keywords
Glivenko-cantelli classesPoincaré propertySet estimation

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal ANNALS OF STATISTICS due to its progression and the good impact it has achieved in recent years, according to the agency WoS (JCR), it has become a reference in its field. In the year of publication of the work, 2014, it was in position 10/122, thus managing to position itself as a Q1 (Primer Cuartil), in the category Statistics & Probability.

From a relative perspective, and based on the normalized impact indicator calculated from the Field Citation Ratio (FCR) of the Dimensions source, it yields a value of: 9.85, which indicates that, compared to works in the same discipline and in the same year of publication, it ranks as a work cited above average. (source consulted: Dimensions May 2025)

Specifically, and according to different indexing agencies, this work has accumulated citations as of 2025-05-02, the following number of citations:

  • WoS: 17
  • Scopus: 18
  • Google Scholar: 34
  • OpenCitations: 18
Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-05-02:

  • The use, from an academic perspective evidenced by the Altmetric agency indicator referring to aggregations made by the personal bibliographic manager Mendeley, gives us a total of: 11.
  • The use of this contribution in bookmarks, code forks, additions to favorite lists for recurrent reading, as well as general views, indicates that someone is using the publication as a basis for their current work. This may be a notable indicator of future more formal and academic citations. This claim is supported by the result of the "Capture" indicator, which yields a total of: 11 (PlumX).

With a more dissemination-oriented intent and targeting more general audiences, we can observe other more global scores such as:

  • The Total Score from Altmetric: 0.25.
  • The number of mentions on the social network X (formerly Twitter): 1 (Altmetric).

It is essential to present evidence supporting full alignment with institutional principles and guidelines on Open Science and the Conservation and Dissemination of Intellectual Heritage. A clear example of this is:

  • The work has been submitted to a journal whose editorial policy allows open Open Access publication.
  • Assignment of a Handle/URN as an identifier within the deposit in the Institutional Repository: https://repositorio.uam.es/handle/10486/669371
Leadership analysis of institutional authors

This work has been carried out with international collaboration, specifically with researchers from: Uruguay.