Item Infomation
Title: | Estimating the probability that a given vector is in the convex hull of a random sample |
Authors: | Satoshi, Hayakawa Terry, Lyons Harald, Oberhauser |
Issue Date: | 2023 |
Publisher: | Springer |
Abstract: | 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. |
Description: | CC BY |
URI: | https://link.springer.com/article/10.1007/s00440-022-01186-1 https://dlib.phenikaa-uni.edu.vn/handle/PNK/7466 |
Appears in Collections | OER - Khoa học Tự nhiên |
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