Signatures of indirect majority systems

2001 ◽  
Vol 38 (2) ◽  
pp. 597-603 ◽  
Author(s):  
Philip J. Boland
Keyword(s):  
Order Statistic ◽  
Coherent System ◽  
Simple Majority ◽  
Distributed Components ◽  
Independent Identically Distributed

If τ is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r × s = n is symmetric around , and use this to show that the expected lifetime of an r × s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.

Signatures of indirect majority systems

2001 ◽  
Vol 38 (02) ◽  
pp. 597-603 ◽  
Author(s):  
Philip J. Boland
Keyword(s):  
Order Statistic ◽  
Coherent System ◽  
Simple Majority ◽  
Distributed Components ◽  
Independent Identically Distributed

If τ is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r × s = n is symmetric around , and use this to show that the expected lifetime of an r × s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals On relative ageing of coherent systems with dependent identically distributed components

2020 ◽  
Vol 52 (1) ◽  
pp. 348-376
Author(s):  
Nil Kamal Hazra ◽  
Neeraj Misra
Keyword(s):  
Sufficient Conditions ◽  
System Level ◽  
Hazard Rates ◽  
Coherent System ◽  
Coherent Systems ◽  
Component Level ◽  
Distributed Components ◽  
Two Systems

AbstractRelative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

ON RELATIVE AGING COMPARISONS OF COHERENT SYSTEMS WITH IDENTICALLY DISTRIBUTED COMPONENTS

2020 ◽  
pp. 1-15 ◽  
Author(s):  
Nil Kamal Hazra ◽  
Neeraj Misra
Keyword(s):  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Orders ◽  
Coherent System ◽  
Coherent Systems ◽  
Cumulative Hazard ◽  
Numerical Examples ◽  
Distributed Components ◽  
Rate Functions ◽  
Important Notion

The relative aging is an important notion which is useful to measure how a system ages relative to another one. Among the existing stochastic orders, there are two important orders describing the relative aging of two systems, namely, aging faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems. Moreover, some numerical examples are given to illustrate the applications of proposed results.

scholarly journals Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

2010 ◽  
Vol 47 (03) ◽  
pp. 876-885 ◽  
Author(s):  
Zhengcheng Zhang
Keyword(s):  
Order Statistics ◽  
Reliability Function ◽  
Coherent System ◽  
Coherent Systems ◽  
Distributed Components ◽  
Inactivity Time ◽  
Independent And Identically Distributed

In this paper we obtain several mixture representations of the reliability function of the inactivity time of a coherent system under the condition that the system has failed at time t (> 0) in terms of the reliability functions of inactivity times of order statistics. Some ordering properties of the inactivity times of coherent systems with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors between systems.

A Note on the Mixture Representation of the Conditional Residual Lifetime of a Coherent System

2013 ◽  
Vol 50 (02) ◽  
pp. 475-485 ◽  
Author(s):  
Xiuying Feng ◽  
Shuhong Zhang ◽  
Xiaohu Li
Keyword(s):  
Order Statistics ◽  
Stochastic Ordering ◽  
Reliability Function ◽  
Residual Lifetime ◽  
Coherent System ◽  
Distributed Components ◽  
Residual Lifetimes ◽  
Independent And Identically Distributed

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

scholarly journals On the Conditional Residual Life and Inactivity Time of Coherent Systems

2014 ◽  
Vol 51 (4) ◽  
pp. 990-998 ◽  
Author(s):  
A. Parvardeh ◽  
N. Balakrishnan
Keyword(s):  
Residual Life ◽  
Coherent System ◽  
Coherent Systems ◽  
Stochastic Comparisons ◽  
Distributed Components ◽  
Inactivity Time ◽  
Independent And Identically Distributed

In this paper we derive mixture representations for the reliability functions of the conditional residual life and inactivity time of a coherent system with n independent and identically distributed components. Based on these mixture representations we carry out stochastic comparisons on the conditional residual life, and the inactivity time of two coherent systems with independent and identical components.

The asymptotic theory of concomitants of order statistics

1974 ◽  
Vol 11 (04) ◽  
pp. 762-770 ◽  
Author(s):  
H. A. David ◽  
J. Galambos
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Random Sample ◽  
Order Statistic ◽  
Asymptotic Theory ◽  
Selection Procedures ◽  
Concomitants Of Order Statistics ◽  
Bivariate Normal ◽  
Independent Identically Distributed

In a random sample of n pairs (X r , Y r ), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr :n be the rth order statistic among the Xr and let Y [r:n] be the Y-variate paired with Xr :n . The Y[r:n] , which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr :n . It is shown that asymptotically the k quantities k fixed, are independent, identically distributed variates. In addition, putting Rt,n for the number of integers j for which , the asymptotic distribution and all moments of n– 1 Rt, n are determined for t such that t/n → λ with 0 < λ < 1.

A characterization of order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously

1985 ◽  
Vol 22 (02) ◽  
pp. 314-323
Author(s):  
A. Deffner ◽  
E. Haeusler
Keyword(s):  
Order Statistic ◽  
Real Line ◽  
Time Scale ◽  
Point Processes ◽  
Present Note ◽  
Poisson Processes ◽  
Scale Transformation ◽  
The Real ◽  
Mixed Sample

The results of Nawrotzki (1962), Feigin (1979) and Puri (1982) show that the class of all point processes (on the real line) with the order statistic property consists of all mixed Poisson processes up to a time-scale transformation, and of all mixed sample processes. The present note characterizes those order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously.

Sign in / Sign up

Export Citation Format

Share Document