Raúl Jiménez

Nonparametric estimation of surface integrals

The estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics, with powerful applications to physics and image analysis, among other fields. Provided that one can determine whether random shots hit the body, Cuevas et al. [Ann. Statist. 35 (2007) 1031―1051] estimate the boundary measure (the boundary length for planar sets and the surface area for 3-dimensional objects) via the consideration of shots at a box containing the body. The statistics considered by these authors, as well as those in subsequent papers, are based on the estimation of Minkowski content and depend on a smoothing parameter which must be carefully chosen. For the same sampling scheme, we introduce a new approach which bypasses this issue, providing strongly consistent estimators of both the boundary measure and the surface integrals of scalar functions, provided one can collect the function values at the sample points. Examples arise in experiments in which the density of the body can be measured by physical properties of the impacts, or in situations where such quantities as temperature and humidity are observed by randomly distributed sensors. Our method is based on random Delaunay triangulations and involves a simple procedure for surface reconstruction from a dense cloud of points inside and outside the body. We obtain basic asymptotics of the estimator, perform simulations and discuss, via Google Earths data, an application to the image analysis of the Aral Sea coast and its cliffs.

The Shared Reward Dilemma

One of the most direct human mechanisms of promoting cooperation is rewarding it. We study the effect of sharing a reward among cooperators in the most stringent form of social dilemma, namely the prisoners dilemma (PD). Specifically, for a group of players that collect payoffs by playing a pairwise PD game with their partners, we consider an external entity that distributes a fixed reward equally among all cooperators. Thus, individuals confront a new dilemma: on the one hand, they may be inclined to choose the shared reward despite the possibility of being exploited by defectors; on the other hand, if too many players do that, cooperators will obtain a poor reward and defectors will outperform them. By appropriately tuning the amount to be shared a vast variety of scenarios arises, including the traditional ones in the study of cooperation as well as more complex situations where unexpected behavior can occur. We provide a complete classification of the equilibria of the n-player game as well as of its evolutionary dynamics.

Asymptotics for Statistical Distances Based on Voronoi Tessellations

We obtain an information-type inequality and a strong law for a wide class of statistical distances between empirical estimates and random measures based on Voronoi tessellations. This extends some basic results in the asymptotic theory of sample spacings, when the cells of the Voronoi tessellation are interpreted as d-dimensional spacings.

Forensic analysis of the Venezuelan recall referendum

The best way to reconcile political actors in a controversial electoral process is a full audit. When this is not possible, statistical tools may be useful for measuring the likelihood of the results. The Venezuelan recall referendum (2004) provides a suitable dataset for thinking about this important problem. The cost of errors in examining an allegation of electoral fraud can be enormous. They can range from legitimizing an unfair election to supporting an unfounded accusation, with serious political implications. For this reason, we must be very selective about data, hypotheses and test statistics that will be used. This article offers a critical review of recent statistical literature on the Venezuelan referendum. In addition, we propose a testing methodology, based exclusively on vote counting, that is potentially useful in election forensics. The referendum is reexamined, offering new and intriguing aspects to previous analyses. The main conclusion is that there were a significant number of irregularities in the vote counting that introduced a bias in favor of the winning option. A plausible scenario in which the irregularities could overturn the results is also discussed.

Statistical Distances Based on Euclidean Graphs

A general approach, based on covering by cells, induced by Euclidean graphs, is developed to provide asymptotic characterizations of multivariate sample densities. This approach provides high dimensional analogs of basic results for random partitions based on one-dimensional sample spacings. The methods used in the proofs yield asymptotics for empirical φ-divergences based on k-spacings and also for the total edge length of the graphs involved.

Strong laws for Euclidean graphs with general edge weights

Consider a random Euclidean graph G with a vertex set consisting of i.i.d. random variables with a common density f. Let the edge lengths e, e[set membership, variant]G, be weighted by a function [phi]. We provide sufficient conditions on G and [phi] guaranteeing that the total edge length functional [summation operator]e[set membership, variant]G [phi](e) satisfies a strong law of large numbers. The limiting constant is shown to depend explicitly on f and [phi].

Probabilistic approach to wave propagation problems, I

In this note we obtain, via probabilistic representation of its Laplace transform, an existence and uniqueness result of solutions to a class of hyperbolic scalar equations. As an interesting consequence of the probabilistic approach, we obtain some high frequency behaviour of the solutions.

Forensic analysis of Venezuelan elections during the Chavez presidency.

Hugo Chávez dominated the Venezuelan electoral landscape since his first presidential victory in 1998 until his death in 2013. Nobody doubts that he always received considerable voter support in the numerous elections held during his mandate. However, the integrity of the electoral system has come into question since the 2004 Presidential Recall Referendum. From then on, different sectors of society have systematically alleged electoral irregularities or biases in favor of the incumbent party. We have carried out a thorough forensic analysis of the national-level Venezuelan electoral processes held during the 1998–2012 period to assess these complaints. The second-digit Benfords law and two statistical models of vote distributions, recently introduced in the literature, are reviewed and used in our case study. In addition, we discuss a new method to detect irregular variations in the electoral roll. The outputs obtained from these election forensic tools are examined taking into account the substantive context of the elections and referenda under study. Thus, we reach two main conclusions. Firstly, all the tools uncover anomalous statistical patterns, which are consistent with election fraud from 2004 onwards. Although our results are not a concluding proof of fraud, they signal the Recall Referendum as a turning point in the integrity of the Venezuelan elections. Secondly, our analysis calls into question the reliability of the electoral register since 2004. In particular, we found irregular variations in the electoral roll that were decisive in winning the 50% majority in the 2004 Referendum and in the 2012 Presidential Elections.

Entropy for Random Partitions and Its Applications

AbstractAsymptotic properties of partitions of the unit interval are studied through the entropy for random partition