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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.

60K10 Applications of renewal theory (reliability, demand theory, etc.)
62E10 Characterization and structure theory of statistical distributions
Full Text: DOI
[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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