A generalization of the maximal-spacings in several dimensions and a convexity test.

Aaron, Catherine; Cholaquidis, Alejandro; Fraiman, Ricardo

doi:10.1007/s10687-017-0284-6

Cited by 7 publications

(25 citation statements)

References 24 publications

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“…In this way, it distinguishes between low and high density regions. Throughout this paper, Janson (1987) calibrated the volume of the maximal spacing under uniformity assumptions without conditions on the shape of the support S. The corresponding extension established in Theorem 2 in Aaron et al (2017) is shown in Theorem 1 modifying slightly the original hypotheses on f and on the shape of S. The result remains true if it is assumed that S is under (R) and the density function f satisfies ( f L 0,1 ):…”

Section: About Maximal Spacingsmentioning

confidence: 91%

“…The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (1983) for uniformly distributed data on the unit cube. Later on, Janson (1987) extended these results to uniformly distributed data on any bounded set and derived the asymptotic distribution of different maximal-spacings notions without conditions on the shape of the support S. Aaron et al (2017) generalized the results by Janson (1987) to the non-uniform case.…”

Section: About Maximal Spacingsmentioning

confidence: 97%

“…Note that the value of the maximal spacing depends on S and also on X n . However, the definition of the maximal inner radius relies only on S. Aaron et al (2017) extended the definition of maximal-spacing assuming that X n is drawn according to a density f with bounded support S, the Lebesgue measure of the set A is one and its barycenter is the origin of R d . In this more general setting, the maximal spacing is defined as…”

Section: About Maximal Spacingsmentioning

confidence: 99%

“…Theorem 1 (Aaron et al (2017)) Let X n be a random and i.i.d sample drawn according to a density f that satisfies ( f L 0,1 ) with compact and nonempty support S under (R). Let U be a random variable with distribution P(U ≤ u) = exp(− exp(−u)) for u ∈ R and let β be a constant specified in Janson (1987).…”

Section: About Maximal Spacingsmentioning

confidence: 99%

“…Recently, Aaron et al (2017) extended the results by Janson (1987) to the case where the data are generated from a density f that is bounded from below and Lipschitz continuous restricted to its bounded support. Here, we will use this extension in order to derive a test to decide, given a fixed r > 0, whether the unknown support S is r −convex with no more information apart from X n .…”

Section: Introductionmentioning

confidence: 97%

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Spatial distribution of invasive species: an extent of occurrence approach

Rodríguez-Casal

Saavedra-Nieves

2021

TEST

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Ecological Risk Assessment faces the challenge of determining the impact of invasive species on biodiversity conservation. Although many statistical methods have emerged in recent years in order to model the evolution of the spatio-temporal distribution of invasive species, the notion of extent of occurrence, formally defined by the International Union for the Conservation of Nature, has not been properly handled. In this work, a novel and flexible reconstruction of the extent of occurrence from occurrence data will be established from nonparametric support estimation theory. Mathematically, given a random sample of points from some unknown distribution, we establish a new data-driven method for estimating its probability support S in general dimension. Under the mild geometric assumption that S is $$r-$$ r - convex, the smallest $$r-$$ r - convex set which contains the sample points is the natural estimator. A stochastic algorithm is proposed for determining an optimal estimate of r from the data under regularity conditions on the density function. The performance of this estimator is studied by reconstructing the extent of occurrence of an assemblage of invasive plant species in the Azores archipelago.

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Section: About Maximal Spacingsmentioning

confidence: 91%

Section: About Maximal Spacingsmentioning

confidence: 97%

Section: About Maximal Spacingsmentioning

confidence: 99%

Section: About Maximal Spacingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Spatial distribution of invasive species: an extent of occurrence approach

Rodríguez-Casal

Saavedra-Nieves

2021

TEST

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Random Euclidean coverage from within

Penrose

2023

Probab. Theory Relat. Fields

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Let $$X_1,X_2, \ldots $$ X 1 , X 2 , … be independent random uniform points in a bounded domain $$A \subset \mathbb {R}^d$$ A ⊂ R d with smooth boundary. Define the coverage threshold$$R_n$$ R n to be the smallest r such that A is covered by the balls of radius r centred on $$X_1,\ldots ,X_n$$ X 1 , … , X n . We obtain the limiting distribution of $$R_n$$ R n and also a strong law of large numbers for $$R_n$$ R n in the large-n limit. For example, if A has volume 1 and perimeter $$|\partial A|$$ | ∂ A | , if $$d=3$$ d = 3 then $$\mathbb {P}[n\pi R_n^3 - \log n - 2 \log (\log n) \le x]$$ P [ n π R n 3 - log n - 2 log ( log n ) ≤ x ] converges to $$\exp (-2^{-4}\pi ^{5/3} |\partial A| e^{-2 x/3})$$ exp ( - 2 - 4 π 5 / 3 | ∂ A | e - 2 x / 3 ) and $$(n \pi R_n^3)/(\log n) \rightarrow 1$$ ( n π R n 3 ) / ( log n ) → 1 almost surely, and if $$d=2$$ d = 2 then $$\mathbb {P}[n \pi R_n^2 - \log n - \log (\log n) \le x]$$ P [ n π R n 2 - log n - log ( log n ) ≤ x ] converges to $$\exp (- e^{-x}- |\partial A|\pi ^{-1/2} e^{-x/2})$$ exp ( - e - x - | ∂ A | π - 1 / 2 e - x / 2 ) . We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform.

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