Asymptotic baseline of the hazard rate function of mixtures

Li, Yulin

doi:10.1017/s0021900200000887

Cited by 10 publications

(10 citation statements)

References 10 publications

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“…, X n . There exist several results in the literature which show, under certain conditions, that the asymptotic behaviour, as t → ∞, of the failure rate function of a mixture is the same as the asymptotic behaviour of the failure rate of the strongest member of the mixture; see Block and Joe (1997), Block et al (2003), Hernandez (2004a), andLi (2005). Thus, we might expect similar results to hold forF in (3.1), although thatF is not a mixture of theF (1:i) , since some of the a i are negative.…”

Section: Results Based On Minimal Signatures and Path Setsmentioning

confidence: 99%

Hazard rate ordering of order statistics and systems

Navarro

Shaked

2006

J. Appl. Probab.

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Let X = (X 1 , X 2 , . . . , X n ) be an exchangeable random vector, and write X (1:i) = min{X 1 , X 2 , . . . , X i }, 1 ≤ i ≤ n. In this paper we obtain conditions under which X (1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

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Section: Results Based On Minimal Signatures and Path Setsmentioning

confidence: 99%

Hazard rate ordering of order statistics and systems

Navarro

Shaked

2006

J. Appl. Probab.

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“…They showed that the mixture failure rate for a family of exponential distributions with parameter ) , [ ∞ ∈ a α converges to the failure rate of the strongest population, which is a in this case. Block et al (1993), Block, Li and Savits (2003) and Li (2005) extended this to a general case.…”

Section: Section 5 Mixture Failure Rate For Large Tmentioning

confidence: 89%

Understanding the shape of the mixture failure rate (with engineering and demographic applications)

Finkelstein

2009

Appl Stoch Models Bus & Ind

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Abstract:Mixtures of distributions are usually effectively used for modeling heterogeneity. It is well known that mixtures of DFR distributions are always DFR. On the other hand, mixtures of IFR distributions can decrease, at least in some intervals of time. As IFR distributions often model lifetimes governed by ageing processes, the operation of mixing can dramatically change the pattern of ageing. Therefore, the study of the shape of the observed (mixture) failure rate in a heterogeneous setting is important in many applications. We study discrete and continuous mixtures, obtain conditions for the mixture failure rate to tend to the failure rate of the strongest populations and describe asymptotic behavior as ∞ → t . Some demographic and engineering examples are considered. The corresponding inverse problem is discussed.

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“…Thus, we shall use the following property given in [18,20]. This result extends Lemma 2.5 in [15]. THEOREM 2.1: Let S be a survival function such that…”

Section: Preliminary Resultsmentioning

confidence: 78%

Tail hazard rate ordering properties of order statistics and coherent systems

Navarro

2007

Naval Research Logistics

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Abstract:We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k-out-of-n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions.

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Asymptotic baseline of the hazard rate function of mixtures

Abstract: In this article, we consider the limit behavior of the hazard rate function of mixture distributions, assuming knowledge of the behavior of each individual distribution. We show that the asymptotic baseline function of the hazard rate function is preserved under mixture.

Cited by 10 publications

References 10 publications

Hazard rate ordering of order statistics and systems

Hazard rate ordering of order statistics and systems

Understanding the shape of the mixture failure rate (with engineering and demographic applications)

Tail hazard rate ordering properties of order statistics and coherent systems

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