Stochastic Comparisons of Nonhomogeneous Processes

Belzunce, Félix; Lillo, Rosa E.; Ruiz, J.; Shaked, Moshe

doi:10.1017/s0269964801152058

Cited by 58 publications

(46 citation statements)

References 18 publications

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“…Therefore, we are able to conclude from Theorem 3.11 in Ref. [2] that SOS are ordered in the multivariate stochastic order under the given assumptions. In particular, X * i:n ≤ ST Y * i:n for i = 1, 2, .…”

Section: Moreover It Is Well Known That If M(t) Is a Mean Residual Lmentioning

confidence: 66%

“…Here the distribution F 2 is used to model the influence of the first component failure on the behavior of the other components. The lifetimes of such components will be represented by X (2) 1 , X (2) 2 , . .…”

Section: Properties Of Coherent Systems Based On Sosmentioning

confidence: 99%

“…If the first failure occurs at time x = X * 1:2 , then the residual lifetime from x of the working component is the same as the residual lifetime of a unit with reliabilitȳ F 2 but with age x, that is, the residual lifetime of the working component (X (2) 1 − x|X * 1:2 = x) has reliability function…”

Section: Ordering Properties For K-out-of-n Systems Based On Sosmentioning

confidence: 99%

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Coherent systems based on sequential order statistics

Navarro

Burkschat

2011

Naval Research Logistics

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Abstract:The sequential order statistics (SOS) are a good way to model the lifetimes of the components in a system when the failure of a component at time t affects the performance of the working components at this age t. In this article, we study properties of the lifetimes of the coherent systems obtained using SOS. Specifically, we obtain a mixture representation based on the signature of the system. This representation is used to obtain stochastic comparisons. To get these comparisons, we obtain some ordering properties for the SOS, which in this context represent the lifetimes of k-out-of-n systems. In particular, we show that they are not necessarily hazard rate ordered.

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Section: Moreover It Is Well Known That If M(t) Is a Mean Residual Lmentioning

confidence: 66%

Section: Properties Of Coherent Systems Based On Sosmentioning

confidence: 99%

Section: Ordering Properties For K-out-of-n Systems Based On Sosmentioning

confidence: 99%

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Coherent systems based on sequential order statistics

Navarro

Burkschat

2011

Naval Research Logistics

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“…In [25] this distribution was used to show that log-convexity of λ F does not imply log-convexity of f .…”

Section: Multivariate Dependence Of Spacings Of Generalized Order Stamentioning

confidence: 99%

Multivariate dependence of spacings of generalized order statistics

Burkschat

2009

Journal of Multivariate Analysis

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Multivariate total positivity Strongly multivariate reverse regular rule Conditionally increasing in sequence Negative and positive orthant dependence Right tail increasing in sequence Increasing failure rate Reversed hazard rate Non-negativity of BLUE Dual generalized order statistics a b s t r a c t Multivariate dependence of spacings of generalized order statistics is studied. It is shown that spacings of generalized order statistics from DFR (IFR) distributions have the CIS (CDS) property. By restricting the choice of the model parameters and strengthening the assumptions on the underlying distribution, stronger dependence relations are established.For instance, if the model parameters are decreasingly ordered and the underlying distribution has a log-convex decreasing (log-concave) hazard rate, then the spacings satisfy the MTP 2 (S-MRR 2 ) property. Some consequences of the results are given. In particular, conditions for non-negativity of the best linear unbiased estimator of the scale parameter in a location-scale family are obtained. By applying a result for dual generalized order statistics, we show that in the particular situation of usual order statistics the assumptions can be weakened.

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“…They have also been touched upon in Belzunce, Lillo, Ruiz, and Shaked (2001). Recently, Lillo et al (2001) have studied in detail four shifted stochastic orders, namely the up likelihood ratio order, the down likelihood ratio order, the up hazard rate order, and the down hazard rate order, that have been obtained starting from the well-known likelihood ratio order and hazard rate order.…”

Section: Introductionmentioning

confidence: 99%

Some new results about shifted hazard and shifted likelihood ratio orders

Aboukalam¹,

Kayid²

2007

imf

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Shifted stochastic orders are a useful tool for establishing interesting inequalities. Two such stochastic orders are the up hazard rate order and the up likelihood ratio order. In this article, we give some results on the preservation of the above orderings between the components under the formation of coherent systems with different structures. Both the cases when components either identically distributed or not necessary identically distributed are discussed. Finally, we provide some applications to Poisson and non-homogeneous Poisson shock models.

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Stochastic Comparisons of Nonhomogeneous Processes

Cited by 58 publications

References 18 publications

Coherent systems based on sequential order statistics

Coherent systems based on sequential order statistics

Multivariate dependence of spacings of generalized order statistics

Some new results about shifted hazard and shifted likelihood ratio orders

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