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Stochastic Precedence and Minima Among Dependent Variables


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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  • Arcones MA, Samaniego FJ (2000) On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint. Ann Statist 28(1):116–150

    MathSciNet  Article  Google Scholar 

  • Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt, Rinehart and Winston, Inc., New York. Probability models, International Series in Decision Processes, Series in Quantitative Methods for Decision Making

    MATH  Google Scholar 

  • Blyth CR (1972) Some probability paradoxes in choice from among random alternatives. J Amer Statist Assoc 67:366–373, 373–381

    MathSciNet  Article  Google Scholar 

  • Blyth CR (1973) Simpson’s paradox and mutually favorable events. J Amer Statist Assoc 68:746

    MathSciNet  Article  Google Scholar 

  • Boland PJ, Singh H, Cukic B (2004) The stochastic precedence ordering with applications in sampling and testing. J Appl Probab 41(1):73–82

    MathSciNet  Article  Google Scholar 

  • Brams SJ, Fishburn PC (1983) Approval voting. Birkhäuser, Boston

    MATH  Google Scholar 

  • Chen R, Zame A (1979) On fair coin-tossing games. J Multivariate Anal 9 (1):150–156

    MathSciNet  Article  Google Scholar 

  • De Santis E, Spizzichino F (2012) First occurrence of a word among the elements of a finite dictionary in random sequences of letters. Electron J Probab 17(25):9

    MathSciNet  MATH  Google Scholar 

  • De Santis E, Spizzichino F (2016a) Some sufficient conditions for stochastic comparisons between hitting times for skip-free Markov chains. Methodol Comput Appl Probab 18(4):1021–1034

    MathSciNet  Article  Google Scholar 

  • De Santis E, Spizzichino F (2016b) Usual and stochastic tail orders between hitting times for two Markov chains. Appl Stoch Models Bus Ind 32(4):526–538

    MathSciNet  Article  Google Scholar 

  • De Santis E, Fantozzi F, Spizzichino F (2015) Relations between stochastic orderings and generalized stochastic precedence. Probab Engrg Inform Sci 29(3):329–343

    MathSciNet  Article  Google Scholar 

  • Durante F, Foschi R (2014) Dependence of exchangeable residual lifetimes subject to failure. Appl Math Comput 235:502–511

    MathSciNet  MATH  Google Scholar 

  • Finkelstein M, Hazra NK, Cha JH (2018) On optimal operational sequence of components in a warm standby system. J Appl Probab 55(4):1014–1024

    MathSciNet  Article  Google Scholar 

  • Fishburn PC (1973) The theory of social choice. Princeton University Press, Princeton

    Book  Google Scholar 

  • Guibas LJ, Odlyzko AM (1981) String overlaps, pattern matching, and nontransitive games. J Combin Theory Ser A 30(2):183–208

    MathSciNet  Article  Google Scholar 

  • Iyer S (1992) The Barlow-Proschan importance and its generalizations with dependent components. Stochastic Process Appl 42(2):353–359

    MathSciNet  Article  Google Scholar 

  • Li S-YR (1980) A martingale approach to the study of occurrence of sequence patterns in repeated experiments. Ann Probab 8(6):1171–1176

    MathSciNet  Article  Google Scholar 

  • Marichal J-L, Mathonet P (2013) On the extensions of Barlow-Proschan importance index and system signature to dependent lifetimes. J Multivariate Anal 115:48–56

    MathSciNet  Article  Google Scholar 

  • Miziuła P, Navarro J (2019) Birnbaum importance measure for reliability systems with dependent components. IEEE Trans Reliab 68(2):439–450

    Article  Google Scholar 

  • Navarro J, Durante F (2017) Copula-based representations for the reliability of the residual lifetimes of coherent systems with dependent components. J Multivariate Anal 158:87–102

    MathSciNet  Article  Google Scholar 

  • Navarro J, Rubio R (2010) Comparisons of coherent systems using stochastic precedence. TEST 19(3):469–486

    MathSciNet  Article  Google Scholar 

  • Norris JR (1998) Markov chains, volume 2 of Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge. Reprint of 1997 original

    Google Scholar 

  • Rychlik T, Spizzichino F (Submitted) Load-sharing reliability models with different component sensibilities on other component working states

  • Saari DG (1995) A chaotic exploration of aggregation paradoxes. SIAM Rev 37 (1):37–52

    MathSciNet  Article  Google Scholar 

  • Savage RP (1994) Jr The paradox of nontransitive dice. Amer Math Monthly 101(5):429–436

    MathSciNet  Article  Google Scholar 

  • Scarsini M, Spizzichino F (1999) Simpson-type paradoxes, dependence, and ageing. J Appl Probab 36(1):119–131

    MathSciNet  Article  Google Scholar 

  • Shaked M, Shanthikumar JG (1990) Dynamic construction and simulation of random vectors. In: Topics in statistical dependence (Somerset, PA, 1987), volume 16 of IMS lecture notes monogr. Ser. Inst. Math. Statist., Hayward, pp 415–433

  • Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Probability and mathematical statistics. Academic Press Inc., Boston

    MATH  Google Scholar 

  • Shaked M, Shanthikumar JG (2015) Multivariate conditional hazard rate functions—an overview. Appl Stoch Models Bus Ind 31(3):285–296

    MathSciNet  Article  Google Scholar 

  • Simpson EH (1951) The interpretation of interaction in contingency tables. J Roy Statist Soc Ser B 13:238–241

    MathSciNet  MATH  Google Scholar 

  • Spizzichino F (2001) Subjective probability models for lifetimes, volume 91 of monographs on statistics and applied probability. Chapman & Hall/CRC, Boca Raton

    Book  Google Scholar 

  • Spizzichino F (2015) Some remarks on multivariate conditional hazard rates and dependence modeling. In: Recent advances in probability and statistics, volume 12 of Lect. Notes Semin. Interdiscip. Mat. Semin. Interdiscip. Mat. (S.I.M.), Potenza, pp 255–271

  • Spizzichino F (2019) Reliability, signature, and relative quality functions of systems under time-homogeneous load-sharing models. Appl Stoch Models Bus Ind 35(2):158–176

    MathSciNet  Article  Google Scholar 

  • Steinhaus H, Trybula S (1959) On a paradox in applied probabilities. Bull Acad Polon Sci 7(67–69):108

    MATH  Google Scholar 

  • Trybula S (1969) Cyclic random inequalities. Zastos Mat 10:123–127

    MathSciNet  MATH  Google Scholar 

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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 23, 187–205 (2021).

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  • 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