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Estimating the probability that a given vector is in the convex hull of a random sample

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Main Authors: Satoshi, Hayakawa, Terry, Lyons, Harald, Oberhauser
Format: Book
Language:English
Published: Springer 2023
Subjects:
d-dimensional random vector X
pn,X(θ)
Online Access:https://link.springer.com/article/10.1007/s00440-022-01186-1
https://dlib.phenikaa-uni.edu.vn/handle/PNK/7466
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spelling oai:localhost:PNK-74662023-04-04T02:08:03Z Estimating the probability that a given vector is in the convex hull of a random sample Satoshi, Hayakawa Terry, Lyons Harald, Oberhauser d-dimensional random vector X pn,X(θ) CC BY For a d-dimensional random vector X, let pn,X(θ) be the probability that the convex hull of n independent copies of X contains a given point θ. We provide several sharp inequalities regarding pn,X(θ) and NX(θ) denoting the smallest n for which pn,X(θ)≥1/2. As a main result, we derive the totally general inequality 1/2≤αX(θ)NX(θ)≤3d+1 , where αX(θ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point θ. We also show several applications of our general results: one is a moment-based bound on NX(E[X]) , which is an important quantity in randomized approaches to cubature construction or measure reduction problem. 2023-04-04T02:08:03Z 2023-04-04T02:08:03Z 2023 Book https://link.springer.com/article/10.1007/s00440-022-01186-1 https://dlib.phenikaa-uni.edu.vn/handle/PNK/7466 en application/pdf Springer
institution Digital Phenikaa
collection Digital Phenikaa
language English
topic d-dimensional random vector X
pn,X(θ)
spellingShingle d-dimensional random vector X
pn,X(θ)
Satoshi, Hayakawa
Terry, Lyons
Harald, Oberhauser
Estimating the probability that a given vector is in the convex hull of a random sample
description CC BY
format Book
author Satoshi, Hayakawa
Terry, Lyons
Harald, Oberhauser
author_facet Satoshi, Hayakawa
Terry, Lyons
Harald, Oberhauser
author_sort Satoshi, Hayakawa
title Estimating the probability that a given vector is in the convex hull of a random sample
title_short Estimating the probability that a given vector is in the convex hull of a random sample
title_full Estimating the probability that a given vector is in the convex hull of a random sample
title_fullStr Estimating the probability that a given vector is in the convex hull of a random sample
title_full_unstemmed Estimating the probability that a given vector is in the convex hull of a random sample
title_sort estimating the probability that a given vector is in the convex hull of a random sample
publisher Springer
publishDate 2023
url https://link.springer.com/article/10.1007/s00440-022-01186-1
https://dlib.phenikaa-uni.edu.vn/handle/PNK/7466
_version_ 1762274897170530304
score 8.887836

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