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Characterizations of the hazard rate order and IFR aging notion. (English) Zbl 1062.60010

The authors show that two random variables \(X\) and \(Y\) satisfy \(X\leq_{\text{hr}}Y\) (that is, are ordered with respect to the hazard rate order) if and only if \([X-t\mid X>t]\leq_{\text{Lt}}[Y-t\mid Y>t]\) (that is, are ordered with respect to the Laplace transform order) for all \(t\). As consequences they obtain new characterizations of the hazard rate order, and of the IFR (increasing failure rate) aging notion, by means of the mean residual life order, and the DMRL (decreasing mean residual life) aging notion.

MSC:

60E15 Inequalities; stochastic orderings
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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References:

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