On boundary estimation

Cuevas, Antonio; Rodríguez-Casal, Alberto

doi:10.1239/aap/1086957575

Cited by 64 publications

(49 citation statements)

References 32 publications

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“…This is fulfilled, under some regularity conditions (see, e,g., Cuevas and Rodríguez-Casal (2004), Theorems 1 and 4) by the Devroye-Wise's (1980) estimator…”

Section: The Case Of Unknown Supportmentioning

confidence: 99%

Testing multivariate uniformity: The distance‐to‐boundary method

Berrendero¹,

Cuevas²,

Vazquez-Grande³

2006

Can J Statistics

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Given a random sample taken on a compact domain S ⊂ R d , the authors propose a new method for testing the hypothesis of uniformity of the underlying distribution. The test statistic is based on the distance of every observation to the boundary of S. The proposed test has a number of interesting properties, namely: Unlike most available methods, it is feasible and particularly suitable for high dimensional data, it is distribution-free for a wide range of choices of S, it can be adapted to the case that the support S is unknown and also allows for one-sided versions. Moreover, the results suggest that, in some cases, this procedure does not suffer from the well-known "curse of dimensionality". The authors study the properties of this test from both a theoretical and practical point of view. In particular, an extensive simulation study is given in order to compare the performance of our methods with some recent alternative procedures. The conclusions suggest that the proposed test provides quite a satisfactory balance between statistical power, computational simplicity, and flexibility of application for different dimensions and supports. Title in French: we can supply thisRésumé : Given a random sample taken on a compact domain S ⊂ R d , the authors propose a new method for testing the hypothesis of uniformity of the underlying distribution. The test statistic is based on the distance of every observation to the boundary of S. The proposed test has a number of interesting properties, namely: Unlike most available methods, it is feasible and particularly suitable for high dimensional data, it is distribution-free for a wide range of choices of S, it can be adapted to the case that the support S is unknown and also allows for one-sided versions. Moreover, the results suggest that, in some cases, this procedure does not suffer from the well-known "curse of dimensionality". The authors study the properties of this test from both a theoretical and practical point of view. In particular, an extensive simulation study is given in order to compare the performance of our methods with some recent alternative procedures. The conclusions suggest that the proposed test provides quite a satisfactory balance between statistical power, computational simplicity, and flexibility of application for different dimensions and supports.

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“…This is fulfilled, under some regularity conditions (see, e,g., Cuevas and Rodríguez-Casal (2004), Theorems 1 and 4) by the Devroye-Wise's (1980) estimator…”

Section: The Case Of Unknown Supportmentioning

confidence: 99%

Testing multivariate uniformity: The distance‐to‐boundary method

Berrendero¹,

Cuevas²,

Vazquez-Grande³

2006

Can J Statistics

Self Cite

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“…The distance d H (X, Y ) is small even if X and Y are quite different in many important aspects. In order to avoid these situations, following Cuevas and Rodríguez-Casal [13] and Cuevas et al [12], a way to reinforce the notion of visual proximity between two sets provided by d H is to impose the proximity of the respective boundaries. We will follow this approach in Theorem 2.1 below.…”

Section: Consistency In Terms Of the Hausdorff Distancementioning

confidence: 99%

“…The estimation of regression level sets in a compact setting has been analyzed in Cavalier [9], Laloë [22]. An alternative approach, based on the geometric properties of the compact support sets, has been presented by Hartigan [20], Cuevas and Fraiman [11], Cuevas and Rodríguez-Casal [13]. The problem of estimating general level sets under compactness assumptions has been discussed by Cuevas et al [12].…”

Section: Introductionmentioning

confidence: 99%

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Bernardino

Laloë²,

Maume-Deschamps

et al. 2013

ESAIM: PS

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“…This paper is a further development of the ideas proposed in [8]; see also [6] and [14] for related approaches. Cuevas and Rodríguez-Casal [7] (see also the references therein) studied the problem of boundary approximation from a nonparametric perspective as well, but they did not provide results on the estimation of boundary measures.…”

Section: For Definitions and Properties) In This Equation Given A Smentioning

confidence: 99%

“…Condition (H1) is quite usual in set estimation theory; see, e.g. [7] and the references therein. It can be seen as a regularity assumption on the shape of the set G c which rules out the existence of sharp peaks in the boundary.…”

Section: Hypothesesmentioning

confidence: 99%

Nonparametric estimation of boundary measures and related functionals: asymptotic results

2009

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We study a nonparametric method for estimating the boundary measure of a compact body G ⊂ R d (the boundary length when d = 2 and the surface area for d = 3) in the case when this measure agrees with the corresponding Minkowski content. The estimator we consider is closely related to the one proposed in Cuevas, Fraiman and Rodríguez-Casal (2007). Our method relies on two sets of random points, drawn inside and outside the set G, with different sampling intensities. Strong consistency and asymptotic normality are obtained under some shape hypotheses on the set G. Some applications and practical aspects are briefly discussed.

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On boundary estimation

Cited by 64 publications

References 32 publications

Testing multivariate uniformity: The distance‐to‐boundary method

Testing multivariate uniformity: The distance‐to‐boundary method

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Nonparametric estimation of boundary measures and related functionals: asymptotic results

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