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Stochastic Precedence and Minima Among Dependent Variables

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Abstract

The notion of stochastic precedence between two random variables emerges as a relevant concept in several fields of applied probability. When one consider a vector of random variables X1,...,Xn, this notion has a preeminent role in the analysis of minima of the type \(\min \limits _{j \in A} X_{j}\) for A ⊂{1,…n}. In such an analysis, however, several apparently controversial aspects can arise (among which phenomena of “non-transitivity”). Here we concentrate attention on vectors of non-negative random variables with absolutely continuous joint distributions, in which a case the set of the multivariate conditional hazard rate (m.c.h.r.) functions can be employed as a convenient method to describe different aspects of stochastic dependence. In terms of the m.c.h.r. functions, we first obtain convenient formulas for the probability distributions of the variables \(\min \limits _{j \in A} X_{j}\) and for the probability of events \(\{X_{i}=\min \limits _{j \in A} X_{j}\}\). Then we detail several aspects of the notion of stochastic precedence. On these bases, we explain some controversial behavior of such variables and give sufficient conditions under which paradoxical aspects can be excluded. On the purpose of stimulating active interest of readers, we present several comments and pertinent examples.

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Acknowledgements

We would like to thank an anonymous Referee for valuable comments and suggestions which, in particular, led us to add Remarks 2, 3, and 6. Most of the results had been presented at the IWAP conference held in Budapest (Hungary), June 2018. E.D.S. and F.S. acknowledge partial support of Ateneo Sapienza Research Projects “Dipendenza,disuguaglianze e approssimazioni in modelli stocastici” (2015), “Processi stocastici: Teoria e applicazioni” (2016), and “Simmetrie e Disuguaglianze in Modelli Stocastici” (2018). Y.M. would like to express his gratitude to coathors for their invitation and support during his visit at Department of Mathematics, Sapienza University of Rome, in January 2018.

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Correspondence to Emilio De Santis.

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De Santis, E., Malinovsky, Y. & Spizzichino, F. Stochastic Precedence and Minima Among Dependent Variables. Methodol Comput Appl Probab (2020). https://doi.org/10.1007/s11009-020-09772-3

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Keywords

  • Multivariate conditional hazard rates
  • Non-transitivity
  • Aggregation/marginalization paradoxes
  • “Small” variables
  • Initially time–homogeneous models
  • Time–homogeneous load sharing models

Mathematics Subject Classification (2010)

  • 60K10
  • 60E15
  • 91B06