Stochastic Precedence and Minima Among Dependent Variables

Emilio De Santis (), Yaakov Malinovsky () and Fabio Spizzichino ()
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Emilio De Santis: University of Rome La Sapienza
Yaakov Malinovsky: University of Maryland
Fabio Spizzichino: University of Rome La Sapienza

Methodology and Computing in Applied Probability, 2021, vol. 23, issue 1, 187-205

Abstract: 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 j ∈ A X j $\min \limits _{j \in A} X_{j}$ 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 j ∈ A X j $\min \limits _{j \in A} X_{j}$ and for the probability of events { X i = min j ∈ A X j } $\{X_{i}=\min \limits _{j \in A} X_{j}\}$ . 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.

Keywords: Multivariate conditional hazard rates; Non-transitivity; Aggregation/marginalization paradoxes; “Small” variables; Initially time–homogeneous models; Time–homogeneous load sharing models; 60K10; 60E15; 91B06 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11009-020-09772-3

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