3,320 research outputs found
Image estimators based on marked bins
The problem of approximating an "image" S in R^d from a random sample of points is considered. If S is included in a grid of square bins, a plausible estimator of S is defined as the union of the "marked" bins (those containing a sample point). We obtain convergence rates for this estimator and study its performance in the approximation of the border of S. The estimation of "digitalized" images is also addressed by using a Vapnik-Chervonenkis approach. The practical aspects of implementation are discussed in some detail, including some technical improvements on the estimator, whose performance is checked through simulated as well as real data examples
Set Estimation Under Biconvexity Restrictions
A set in the Euclidean plane is said to be biconvex if, for some angle
, all its sections along straight lines with inclination
angles and are convex sets (i.e, empty sets or
segments). Biconvexity is a natural notion with some useful applications in
optimization theory. It has also be independently used, under the name of
"rectilinear convexity", in computational geometry. We are concerned here with
the problem of asymptotically reconstructing (or estimating) a biconvex set
from a random sample of points drawn on . By analogy with the classical
convex case, one would like to define the "biconvex hull" of the sample points
as a natural estimator for . However, as previously pointed out by several
authors, the notion of "hull" for a given set (understood as the "minimal"
set including and having the required property) has no obvious, useful
translation to the biconvex case. This is in sharp contrast with the well-known
elementary definition of convex hull. Thus, we have selected the most commonly
accepted notion of "biconvex hull" (often called "rectilinear convex hull"): we
first provide additional motivations for this definition, proving some useful
relations with other convexity-related notions. Then, we prove some results
concerning the consistent approximation of a biconvex set and and the
corresponding biconvex hull. An analogous result is also provided for the
boundaries. A method to approximate, from a sample of points on , the
biconvexity angle is also given
On the estimation of the influence curve
We prove the asymptotic validity of bootstrap confidence bands for the influence curve from its usual estimator (the sensitive curve). The proof is based on the use of Gill's (1989) generalized delta method for Hadamard differentiable operators. The scope and applicability of this result are also discussed
On robustness properties of bootstrap approximations
Bootstrap approximations to the sampling distribution can be seen as generalized statistics taking values in a space of probability measures. We first analyze qualitative robustness [in Hampel's (1971) sense] of these statistics when the initial estimators {Tn } (whose distributions we want to approximate using bootstrap resampling) are obtained by restriction from a statistical functional T defined for all probability distributions. Whereas continuity of T turns out to be the natural condition to ensure qualitative robustness of {Tn }, we show that the uniform continuity of T is a sufficient condition for robustness of the bootstrap. This result applies to M-estimators.
Next, we study asymptotic properties of the bootstrap estimator for the infiuence function T'(F; x) of T at a distribution F and we prove that continuous Hadamard differentiability of the operator F_ T'(F;.) with respect to F is a natural condition to establish the validity of bootstrap confidence bands for this estimator
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.Comment: 30 pages, 6 figures, 2 table
On the estimation of the influence curve.
We prove the asymptotic validity of bootstrap confidence bands for the influence curve from its usual estimator (the sensitive curve). The proof is based on the use of Gill's (1989) generalized delta method for Hadamard differentiable operators. The scope and applicability of this result are also discussed.Influence curve; Sensitivity curve; Bootstrap confidence bands; Hadamard differentiability;
Revisiting the problem of a crack impinging on an interface: A modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model
ArtÃculo Open Access en el sitio web del editor. Pago por publicar en abierto.The problem of a crack impinging on an interface has been thoroughly investigated in the last three decades due to its important role in the mechanics and physics of solids. In the current investigation, this problem is revisited in view of the recent progresses on the phase field approach of brittle fracture. In this concern, a novel formulation combining the phase field approach for modeling brittle fracture in the bulk and a cohesive zone model for pre-existing adhesive interfaces is herein proposed to investigate the competition between crack penetration and deflection at an interface. The model, implemented within the finite element method framework using a monolithic fully implicit solution strategy, is applied to provide a further insight into the understanding of the role of model parameters on the above competition. In particular, in this study, the role of the fracture toughness ratio between the interface and the adjoining bulks and of the characteristic fracture-length scales of the dissipative models is analyzed. In the case of a brittle interface, the asymptotic predictions based on linear elastic fracture mechanics criteria for crack penetration, single deflection or double deflection are fully captured by the present method. Moreover, by increasing the size of the process zone along the interface, or by varying the internal length scale of the phase field model, new complex phenomena are emerging, such as simultaneous crack penetration and deflection and the transition from single crack penetration to deflection and penetration with subsequent branching into the bulk. The obtained computational trends are in very good agreement with previous experimental observations and the theoretical considerations on the competition and interplay between both fracture mechanics models open new research perspectives for the simulation and understanding of complex fracture patterns.Unión Europea FP/2007-2013/ERC 306622Ministerio de EconomÃa y Competitividad DPI2012-37187, MAT2015-71036-P y MAT2015-71309-PJunta de AndalucÃa P11-TEP-7093 y P12-TEP- 105
A nonparametric approach to the estimation of lengths and surface areas
The Minkowski content of a body represents
the boundary length (for ) or the surface area (for ) of . A
method for estimating is proposed. It relies on a nonparametric
estimator based on the information provided by a random sample (taken on a
rectangle containing ) in which we are able to identify whether every point
is inside or outside . Some theoretical properties concerning strong
consistency, -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.Comment: Published at http://dx.doi.org/10.1214/009053606000001532 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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