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A phase transition for the probability of being a maximum among random vectors with general iid coordinates

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  • Jacobovic, Royi
  • Zuk, Or

Abstract

Consider n iid real-valued random vectors of size k having iid coordinates with a general distribution function F. A vector is a maximum if and only if there is no other vector in the sample that weakly dominates it in all coordinates. Let pk,n be the probability that the first vector is a maximum. The main result of the present paper is that if k≡kn grows at a slower (faster) rate than a certain factor of log(n), then pk,n→0 (resp. pk,n→1) as n→∞. Furthermore, the factor is fully characterized as a functional of F. We also study the effect of F on pk,n, showing that while pk,n may be highly affected by the choice of F, the phase transition is the same for all distribution functions up to a constant factor.

Suggested Citation

  • Jacobovic, Royi & Zuk, Or, 2023. "A phase transition for the probability of being a maximum among random vectors with general iid coordinates," Statistics & Probability Letters, Elsevier, vol. 199(C).
    • Handle: RePEc:eee:stapro:v:199:y:2023:i:c:s0167715223000718
    DOI: 10.1016/j.spl.2023.109847
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    References listed on IDEAS

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    1. Gérard Biau & Erwan Scornet, 2016. "Rejoinder on: A random forest guided tour," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 264-268, June.
    2. Gérard Biau & Erwan Scornet, 2016. "A random forest guided tour," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 197-227, June.
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