# Generalization of the pairwise stochastic precedence order to the sequence of random variables

01 Jul 2021-Probability in the Engineering and Informational Sciences (Cambridge University Press)-Vol. 35, Iss: 3, pp 699-707

TL;DR: In this paper, a new stochastic ordering for the sequence of independent random variables is proposed, the sequential precedence order (SPO), which generalizes the SPO for two random variables to the case n > 2.

Abstract: We discuss a new stochastic ordering for the sequence of independent random variables. It generalizes the stochastic precedence (SP) order that is defined for two random variables to the case n > 2. All conventional stochastic orders are transitive, whereas the SP order is not. Therefore, a new approach to compare the sequence of random variables had to be developed that resulted in the notion of the sequential precedence order. A sufficient condition for this order is derived and some examples are considered.

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TL;DR: This paper focuses attention on ranking schemes associated to m-tuples of non-negative random variables and suitably single out a special subclass of load-sharing models and shows that all possible ranking schemes can be conveniently obtained by only considering such a special family of survival models.

Abstract: In this paper we present a study about minima among random variables, about the context of voting theory, and about paradoxes related with such topics. In the field of reliability theory, the term load-sharing model is commonly used to designate a special type of multivariate survival models. We demonstrate the effectiveness that such dependence models can have also in some other fields, such as those of interest here. Several important, and by now classic, papers have been devoted to single out and to prove general conclusions in the field of voting theory. We reformulate and achieve such conclusions by developing a method of proof, alternative to the existing ones, and completely probabilistic in nature. As main features of this method, we focus attention on ranking schemes associated to m-tuples of non-negative random variables and suitably single out a special subclass of load-sharing models. Then we show that all possible ranking schemes can be conveniently obtained by only considering such a special family of survival models. This result leads to some new insight about the construction of voting situations which give rise to all possible types of voting paradoxes. Our method and related implications will be also illustrated by means of some examples and informative remarks.

3 citations

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TL;DR: In this article , the authors show that load sharing models can be used to obtain basic results about a multivariate extension of stochastic precedence and related paradoxes, which can be applied in several different fields.

Abstract: Abstract We show that load-sharing models (a very special class of multivariate probability models for nonnegative random variables) can be used to obtain basic results about a multivariate extension of stochastic precedence and related paradoxes. Such results can be applied in several different fields. In particular, applications of them can be developed in the context of paradoxes which arise in voting theory. Also, an application to the notion of probability signature may be of interest, in the field of systems reliability.

2 citations

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01 Jun 1981

TL;DR: A number of new classes of life distributions arising naturally in reliability models are treated systematically and each provides a realistic probabilistic description of a physical property occurring in the reliability context, thus permitting more realistic modeling of commonly occurring reliability situations.

Abstract: : This is the first of two books on the statistical theory of reliability and life testing. The present book concentrates on probabilistic aspects of reliability theory, while the forthcoming book will focus on inferential aspects of reliability and life testing, applying the probabilistic tools developed in this volume. This book emphasizes the newer, research aspects of reliability theory. The concept of a coherent system serves as a unifying theme for much of the book. A number of new classes of life distributions arising naturally in reliability models are treated systematically: the increasing failure rate average, new better than used, decreasing mean residual life, and other classes of distributions. As the names would seem to indicate, each such class of life distributions provides a realistic probabilistic description of a physical property occurring in the reliability context. Also various types of positive dependence among random variables are considered, thus permitting more realistic modeling of commonly occurring reliability situations.

3,876 citations

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17 Oct 2007

TL;DR: This paper presents a meta-analysis of the application of Signature-Based Closure, Preservation and Characterization Theorems to Network Reliability and its applications in Reliability Economics and Signature-based Analysis of System Lifetimes.

Abstract: Background on Coherent Systems.- System Signatures.- Signature-Based Closure, Preservation and Characterization Theorems.- Further Signature-Based Analysis of System Lifetimes.- Applications of Signatures to Network Reliability.- Applications of Signatures in Reliability Economics.- Summary and Discussion.

387 citations

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TL;DR: The problem of choosing from among possible random losses or payoffs has been studied in this paper, as in choosing from possible statistical decision procedures or from possible wagers, and the difficulties of choosing among possible losses or wagers have been discussed.

Abstract: The probability P(X>Y) can be arbitrarily close to 1 even though the random variable X is stochastically smaller than Yi the probabilities P(X

122 citations