scholarly journals Preservation of some partial orderings under Poisson shock models

1989 ◽  
Vol 21 (3) ◽  
pp. 713-716 ◽  
Author(s):  
Harshinder Singh ◽  
Kanchan Jain
Keyword(s):  
Poisson Process ◽  
Residual Life ◽  
Stochastic Ordering ◽  
Mean Residual Life ◽  
Survival Functions ◽  
Constant Intensity ◽  
Variable Ordering ◽  
Shock Models ◽  
Partial Orderings ◽  
Likelihood Ratio Ordering

Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let Pk denote the probability that the first device will survive the first k shocks and let denote such a probability for the second device. Let and denote the survival functions of the first and the second device respectively. In this note we show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities and are preserved by the corresponding survival functions and .

scholarly journals Preservation of some partial orderings under Poisson shock models

1989 ◽  
Vol 21 (03) ◽  
pp. 713-716 ◽  
Author(s):  
Harshinder Singh ◽  
Kanchan Jain
Keyword(s):  
Poisson Process ◽  
Residual Life ◽  
Stochastic Ordering ◽  
Mean Residual Life ◽  
Survival Functions ◽  
Constant Intensity ◽  
Variable Ordering ◽  
Shock Models ◽  
Partial Orderings ◽  
Likelihood Ratio Ordering

Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let Pk denote the probability that the first device will survive the first k shocks and let denote such a probability for the second device. Let and denote the survival functions of the first and the second device respectively. In this note we show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities and are preserved by the corresponding survival functions and .

Shock models leading to increasing failure rate and decreasing mean residual life survival

1982 ◽  
Vol 19 (01) ◽  
pp. 158-166 ◽  
Author(s):  
Malay Ghosh ◽  
Nader Ebrahimi
Keyword(s):  
Poisson Process ◽  
Failure Rate ◽  
Residual Life ◽  
Mean Residual Life ◽  
Increasing Failure Rate ◽  
Shock Models

Shock models leading to various univariate and bivariate increasing failure rate (IFR) and decreasing mean residual life (DMRL) distributions are discussed. For proving the IFR properties, shocks are not necessarily assumed to be governed by a Poisson process.

Shock models leading to increasing failure rate and decreasing mean residual life survival

1982 ◽  
Vol 19 (1) ◽  
pp. 158-166 ◽  
Author(s):  
Malay Ghosh ◽  
Nader Ebrahimi
Keyword(s):  
Poisson Process ◽  
Failure Rate ◽  
Residual Life ◽  
Mean Residual Life ◽  
Increasing Failure Rate ◽  
Shock Models

Shock models leading to various univariate and bivariate increasing failure rate (IFR) and decreasing mean residual life (DMRL) distributions are discussed. For proving the IFR properties, shocks are not necessarily assumed to be governed by a Poisson process.

Dynamic multivariate mean residual life functions

1991 ◽  
Vol 28 (03) ◽  
pp. 613-629 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Hazard Rate ◽  
Residual Life ◽  
Stochastic Ordering ◽  
Partial Ordering ◽  
Mean Residual Life ◽  
Positive Dependence ◽  
Basic Properties ◽  
Rate Functions ◽  
Hazard Rate Ordering ◽  
Weak Notion

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied. A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given. Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.

A generalized bivariate exponential distribution

1967 ◽  
Vol 4 (2) ◽  
pp. 291-302 ◽  
Author(s):  
Albert W. Marshall ◽  
Ingram Olkin
Keyword(s):  
Poisson Process ◽  
Exponential Distribution ◽  
Residual Life ◽  
Waiting Times ◽  
Practical Importance ◽  
Moment Generating Function ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential ◽  
Shock Models ◽  
Bivariate Poisson Process

In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age.The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.

scholarly journals On preservation of some partial orderings under shock models

1990 ◽  
Vol 22 (02) ◽  
pp. 508-509 ◽  
Author(s):  
Subhash C. Kochar
Keyword(s):  
Poisson Process ◽  
Homogeneous Poisson Process ◽  
Shock Models ◽  
Partial Orderings ◽  
Life Distributions

Singh and Jain (1989) have proved some preservation results for partial orderings of life distributions assuming that shocks occur according to a homogeneous Poisson process. It is shown that their results hold under less restrictive conditions.

Order-Preserving Shock Models

1994 ◽  
Vol 8 (1) ◽  
pp. 125-134 ◽  
Author(s):  
Y. Kebir
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Ordering ◽  
Shock Models ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering

To date, research in shock models has been primarily concerned with the classpreserving properties of certain shock and damage kernels. This article focuses on the order-preserving properties of those kernels. We show that they preserve stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

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