MULTIVARIATE STOCHASTIC COMPARISONS OF SEQUENTIAL ORDER STATISTICS

2006 ◽  
Vol 21 (1) ◽  
pp. 47-66 ◽  
Author(s):  
Weiwei Zhuang ◽  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Distribution Functions ◽  
Stochastic Orders ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Underlying Distribution ◽  
Sequential Order Statistics ◽  
Multivariate Stochastic Orders ◽  
Multivariate Likelihood

In this article we investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics in the multivariate likelihood ratio, the multivariate hazard rate, and the usual multivariate stochastic orders. Some applications of the main results are also given.

STOCHASTIC COMPARISONS OF SYSTEMS BASED ON SEQUENTIAL ORDER STATISTICS VIA PROPERTIES OF DISTORTED DISTRIBUTIONS

2017 ◽  
Vol 32 (2) ◽  
pp. 246-274 ◽  
Author(s):  
M. Burkschat ◽  
J. Navarro
Keyword(s):  
Order Statistics ◽  
Residual Life ◽  
Stochastic Orders ◽  
Mean Residual Life ◽  
Model Parameters ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Sequential Order Statistics ◽  
Distorted Distributions ◽  
Life Function

We consider systems based on sequential order statistics (SOS) with underlying distributions possessing proportional hazard rates (PHRs). In that case, the lifetime distribution of the system can be expressed as a distorted distribution. Motivated by the distribution structure in the case of pairwise different model parameters, a particular class of distorted distributions, the generalized PHR model, is introduced and characterizations of stochastic comparisons for several stochastic orders are obtained. Moreover, results on the asymptotic behavior of some aging characteristics, for example, the hazard rate and the mean residual life function, of general distorted distributions as well as related bounds are given. The results are supplemented with limiting properties of the systems in the case of possibly equal model parameters. Some examples are presented in order to illustrate the application of the findings to systems based on SOS and also to systems with independent heterogeneous components.

Representations of component importance for coherent systems with exchangeable components

2020 ◽  
Vol 57 (2) ◽  
pp. 385-406
Author(s):  
S. Pitzen ◽  
M. Burkschat
Keyword(s):  
Order Statistics ◽  
Distribution Functions ◽  
Importance Measure ◽  
Coherent Systems ◽  
Sequential Order ◽  
Limiting Behavior ◽  
Sequential Order Statistics ◽  
Component Importance ◽  
Dependent Component ◽  
Two Measures

AbstractTwo definitions of Birnbaum’s importance measure for coherent systems are studied in the case of exchangeable components. Representations of these measures in terms of distribution functions of the ordered component lifetimes are given. As an example, coherent systems with failure-dependent component lifetimes based on the notion of sequential order statistics are considered. Furthermore, it is shown that the two measures are ordered in the case of associated component lifetimes. Finally, the limiting behavior of the measures with respect to time is examined.

scholarly journals Dynamic Signatures of Coherent Systems Based on Sequential Order Statistics

2013 ◽  
Vol 50 (1) ◽  
pp. 272-287 ◽  
Author(s):  
M. Burkschat ◽  
J. Navarro
Keyword(s):  
Order Statistics ◽  
Failure Time ◽  
Residual Lifetime ◽  
Coherent Systems ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Additional Information ◽  
Structured Systems ◽  
Sequential Order Statistics ◽  
Inactivity Time

Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.

Dynamic Signatures of Coherent Systems Based on Sequential Order Statistics

2013 ◽  
Vol 50 (01) ◽  
pp. 272-287 ◽  
Author(s):  
M. Burkschat ◽  
J. Navarro
Keyword(s):  
Order Statistics ◽  
Failure Time ◽  
Residual Lifetime ◽  
Coherent Systems ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Additional Information ◽  
Structured Systems ◽  
Sequential Order Statistics ◽  
Inactivity Time

Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.

STOCHASTIC ORDERINGS BETWEEN SPACINGS OF GENERALIZED ORDER STATISTICS

2002 ◽  
Vol 16 (4) ◽  
pp. 471-484 ◽  
Author(s):  
Manuel Franco ◽  
José M. Ruiz ◽  
M. Carmen Ruiz
Keyword(s):  
Order Statistics ◽  
Record Values ◽  
Generalized Order Statistics ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Sequential Order Statistics ◽  
Normalized Spacings ◽  
Generalized Order

In this article, we establish stochastic comparisons between normalized spacings of generalized order statistics. These comparisons allow us to extend and unify some results obtained by other authors for ordinary order statistics and record values. Furthermore, we can compare spacings of different models (i.e., between ordinary order statistics and sequential order statistics, record values and Pfeifer's record values, and so forth).

Reliability modeling of coherent systems with shared components based on sequential order statistics

2018 ◽  
Vol 55 (3) ◽  
pp. 845-861
Author(s):  
S. Ashrafi ◽  
S. Zarezadeh ◽  
M. Asadi
Keyword(s):  
Order Statistics ◽  
Reliability Function ◽  
Failure Model ◽  
Coherent Systems ◽  
Sequential Order ◽  
Birth Process ◽  
Reliability Modeling ◽  
Joint Reliability ◽  
Sequential Order Statistics ◽  
Signature Matrix

Abstract In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.

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