arXiv subjects

Antonio Cuevas

Publications and source records attributed to Antonio Cuevas.

23 records · Page 2Linked to original sources

arXiv2014

On Poincaré cone property

A domain $S\subset{\mathbb{R}}^d$ 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 $\bar{S}$. 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 $\mathbb{R}^d$. 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.

math.ST↗

arXiv2011

On statistical properties of sets fulfilling rolling-type conditions

Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity and rolling condition. First, the relations between these shape conditions are analyzed. Second, we obtain for the estimation of sets fulfilling a rolling condition a result of "full consistency" (i.e., consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P-uniformity class (in Billingsley and Topsoe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r-convex hull of the sample is proved to be a fully consistent estimator of an r-convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (a.s.) to that of the underlying support. Fifth, the above results are applied to get new consistency statements for level set estimators based on the excess mass methodology (Polonik, 1995).

math.ST↗

arXiv2010

Supervised classification for a family of Gaussian functional models

In the framework of supervised classification (discrimination) for functional data, it is shown that the optimal classification rule can be explicitly obtained for a class of Gaussian processes with "triangular" covariance functions. This explicit knowledge has two practical consequences. First, the consistency of the well-known nearest neighbors classifier (which is not guaranteed in the problems with functional data) is established for the indicated class of processes. Second, and more important, parametric and nonparametric plug-in classifiers can be obtained by estimating the unknown elements in the optimal rule. The performance of these new plug-in classifiers is checked, with positive results, through a simulation study and a real data example.

stat.ML↗

arXiv2008

Supervised functional classification: A theoretical remark and some comparisons

The problem of supervised classification (or discrimination) with functional data is considered, with a special interest on the popular k-nearest neighbors (k-NN) classifier. First, relying on a recent result by Cerou and Guyader (2006), we prove the consistency of the k-NN classifier for functional data whose distribution belongs to a broad family of Gaussian processes with triangular covariance functions. Second, on a more practical side, we check the behavior of the k-NN method when compared with a few other functional classifiers. This is carried out through a small simulation study and the analysis of several real functional data sets. While no global "uniform" winner emerges from such comparisons, the overall performance of the k-NN method, together with its sound intuitive motivation and relative simplicity, suggests that it could represent a reasonable benchmark for the classification problem with functional data.

stat.ML↗

arXiv2007

A nonparametric approach to the estimation of lengths and surface areas

The Minkowski content $L_0(G)$ of a body $G\subset{\mathbb{R}}^d$ represents the boundary length (for $d=2$) or the surface area (for $d=3$) of $G$. A method for estimating $L_0(G)$ is proposed. It relies on a nonparametric estimator based on the information provided by a random sample (taken on a rectangle containing $G$) in which we are able to identify whether every point is inside or outside $G$. Some theoretical properties concerning strong consistency, $L_1$-error and convergence rates are obtained. A practical application to a problem of image analysis in cardiology is discussed in some detail. A brief simulation study is provided.

math.ST↗