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.

Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disease causing death of the motor neurons. Proteotoxicity caused by TDP-43 protein is an important aspect of ALS pathogenesis, with TDP-43 being the main constituent of the aggregates found in patients. We have previously tested the effect of different sugars on the proteotoxicity caused by the expression of mutant TDP-43 in Caenorhabditis elegans. Here we tested maple syrup, a natural compound containing many active molecules including sugars and phenols, for neuroprotective activity. Maple syrup decreased several age-dependent phenotypes caused by the expression of TDP-43(A315T) in C. elegans motor neurons and requires the FOXO transcription factor DAF-16 to be effective.

In this paper we introduce a new estimator for the support of a multivariate density. It is defined as a union of convex hulls of observations contained in balls of fixed radius. We study the asymptotic behavior of this "local convex hull" for the estimation of the support and its boundary. When the support is smooth enough, the proposed estimator is proved to be, eventually almost surely, homeomorphic to the support. Numerical simulations on both simulated and real data illustrate the performance of our estimator.