Stochastic precedence and minima among dependent variables.
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Date
2020-02-14
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Citation of Original Publication
De Santis, Emilio, Yaakov Malinovsky, and Fabio Spizzichino. “Stochastic Precedence and Minima Among Dependent Variables.” Methodology and Computing in Applied Probability 23, no. 1 (March 1, 2021): 187–205. https://doi.org/10.1007/s11009-020-09772-3.
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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 minj∈A Xj for A ⊂ {1, . . . n}. In such an analysis, however,
several apparently controversial aspects can arise (among which phenomena of “nontransitivity”). 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 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
minj∈A Xj and for the probability of events {Xi = minj∈A Xj}. 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.