This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set S ⊂ R d . The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of S, of geometric or topological character. The available information is just a random sample of points drawn on S. The term "to identify" means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: is S full dimensional? Is S "close to a lower dimensional set" M? If so, can we estimate M or some functionals of M (in particular, the Minkowski content of M)? As an important auxiliary tool in the answers of these questions, a denoising procedure is proposed in order to partially remove the noise in the original data. The theoretical results are complemented with some simulations and graphical illustrations. arXiv:1702.05193v2 [math.ST] 3 Nov 2017 Estimation of some other relevant quantities in a manifold, Niyogi, Smale and Weinberger (2008), Chen and Müller (2012). Dimensionality reduction, Genovese et al. (2012a), Tenebaum et al. (2000).The problems under study. The contents of the paper. We are interested in getting some information (in particular, regarding dimensionality and Minkowski content) about a compact set M ⊂ R d . While the set M is typically unknown, we are supposed to have a random sample of points X 1 , . . . , X n whose distribution P X has a support "close to M". To be more specific, we consider two different models:The noiseless model : the support of P X is M itself; Aamari and Levrard (2015), Amenta et al. (2002), Cholaquidis et al. (2014), Cuevas and Fraiman (1997). The parallel (noisy) model : the support of P X is the parallel set S of points within a distance to M smaller than R 1 , for some R 1 > 0, where M is a d -dimensional set and d ≤ d; Berrendero et al. (2014). Note that other different models "with noise" are considered in Genovese et al. (2012a), Genovese et al. (2012b) and Genovese et al (2012c).In Section 3 we first develop, under the noiseless model, an algorithmic procedure to identify, eventually, almost surely (a.s.), whether or not M has an empty interior; this is achieved in Theorems 1 and 2 below. A positive answer would essentially entail (under some conditions, see the beginning of Section 3) that M has a dimension smaller than that of the ambient space.Then, assuming the noisy model andM = ∅ ( whereM denotes the interior of M) Theorems 3 (i) and 4 (i) provide two methods for the estimation of the maximum level of noise R 1 , giving also the corresponding convergence rates. If R 1 is known in advance, the remaining results in Theorems 3 and 4 allow us also to decide whether or not the "inside set" M has an empty interior.The identification methods are "algorithmic" in the sense that they are based on automatic procedures to perform them with arbitrary precision. This will requi...

The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (1983) for data uniformly distributed on the unit cube. Later on, Janson (1987) extended the results to data uniformly distributed on any bounded set, and obtained a very fine result, namely, he derived the asymptotic distribution of different maximal-spacings notions. These results have been very useful in many statistical applications.We extend Janson's results to the case where the data are generated from a Hölder continuous density that is bounded from below and whose support is bounded. As an application, we develop a convexity test for the support of a distribution.

We introduce a nonlinear aggregation type classifier for functional data defined on a separable and complete metric space. The new rule is built up from a collection of M arbitrary training classifiers. If the classifiers are consistent, then so is the aggregation rule. Moreover, asymptotically the aggregation rule behaves as well as the best of the M classifiers. The results of a small simulation are reported both, for high dimensional and functional data.

We study the problem of estimating a compact set S ⊂ R d from a trajectory of a reflected Brownian motion in S with reflections on the boundary of S. We establish consistency and rates of convergence for various estimators of S and its boundary. This problem has relevant applications in ecology in estimating the home range of an animal based on tracking data. There are a variety of studies on the habitat of animals that employ the notion of home range. This paper offers theoretical foundations for a new methodology that, under fairly unrestrictive shape assumptions, allows one to find flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.

Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in: AbstractThe general aim of manifold estimation is reconstructing, by statistical methods, an m-dimensional compact manifold S on R d (with m ≤ d) or estimating some relevant quantities related to the geometric properties of S. We will assume that the sample data are given by the distances to the (d − 1)-dimensional manifold S from points randomly chosen on a band surrounding S, with d = 2 and d = 3. The point in this paper is to show that, if S belongs to a wide class of compact sets (which we call sets with polynomial volume), the proposed statistical model leads to a relatively simple parametric formulation. In this setup, standard methodologies (method of moments, maximum likelihood) can be used to estimate some interesting geometric parameters, including curvatures and Euler characteristic. We will particularly focus on the estimation of the (d − 1)-dimensional boundary measure (in Minkowski's sense) of S.It turns out, however, that the estimation problem is not straightforward since the standard estimators show a remarkably pathological behavior: while they are consistent and asymptotically normal, their expectations are infinite. The theoretical and practical consequences of this fact are discussed in some detail.