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Characterizations of the IFR and DFR aging notions by means of the dispersive order. (English) Zbl 0903.60081
Summary: The purpose of this note is to show a new characterization of the IFR and DFR aging notions by means of the dispersive order. Our result extends results of L. Mailhot [C. R. Acad. Sci., Paris, Sér. I 304, 499-501 (1987; Zbl 0617.60015)] and of F. Belzunce, J. Candel and J. M. Ruiz [Stat. Probab. Lett. 28, No. 4, 321-327 (1996; Zbl 0854.62011)]. Our method of proof can also be extended to handle comparisons of “past lives”, that is, comparisons of $$[X- t\mid X<t]$$ for all $$t$$, and characterizations of random variables with monotone reversed hazard rates.

##### MSC:
 60K10 Applications of renewal theory (reliability, demand theory, etc.) 62E10 Characterization and structure theory of statistical distributions
##### Keywords:
aging; stochastic orders
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##### References:
 [1] Belzunce, F.; Candel, J.; Ruiz, J.M., Dispersive ordering and characterizations of ageing classes, () · Zbl 0854.62011 [2] Cao, J.; Wang, Y., The NBUC and NWUC classes of life distributions, J. appl. probab., 28, 473-479, (1991) · Zbl 0729.62096 [3] Mailhot, L., Ordre de dispersion et lois tronquées, C.R. acad. sc. Paris, t. 304, Série I, 16, 499-501, (1987) · Zbl 0617.60015 [4] Shaked, M.; Shanthikumar, J.G., Characterizations of some first passage times using log-concavity and log-convexity as aging notions, Probab. eng. inform. sci., 1, 279-291, (1987) · Zbl 1133.60309 [5] Shaked, M.; Shanthikumar, J.G., Stochastic orders and their applications, (1994), Academic Press New York · Zbl 0806.62009
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