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Dynamic multivariate aging notions in relability theory. (English) Zbl 0732.62094
This paper uses multivariate extensions of likelihood ratio, hazard rate and stochastic orderings to suggest a variety of possible definitions for increasing failure rate (IFR) and Polya frequency of order 2 \((PF_ 2)\) for random vectors. Logical relationships amongst the definitions are considered. Single classes of multivariate IFR and \(PF_ 2\) distributions emerge, which are recommended on theoretical and practical grounds. The former class is included in that introduced by E. Arjas [Math. Oper. Res. 6, 551-562 (1981; Zbl 0501.60091)].
Reviewer: J.Preater (Keele)

62N05 Reliability and life testing
62H05 Characterization and structure theory for multivariate probability distributions; copulas
90B25 Reliability, availability, maintenance, inspection in operations research
Full Text: DOI
[1] Arjas, E., The failure and hazard processes in multivariate reliability systems, Math. oper. res., 6, 551-562, (1981) · Zbl 0501.60091
[2] Arjas, E.; Norros, I., Life lengths and association: a dynamic approach, Math. oper. res., 9, 151-158, (1984) · Zbl 0531.90041
[3] Freund, J.E., A bivariate extension of the exponential distribution, J. amer. statist. assoc., 56, 971-977, (1961) · Zbl 0106.13304
[4] Kamae, T.; Krengel, U.; O’Brien, G.L., Stochastic inequalities on partially ordered spaces, Ann. probab., 5, 899-912, (1977) · Zbl 0371.60013
[5] Karlin, S.; Rinott, Y., Classes of orderings of measures and related correlation inequalities. I. multivariate totally positive distributions, J. multivariate anal., 10, 467-498, (1980) · Zbl 0469.60006
[6] Norros, I., Systems weakened by failures, Stochastic process. appl., 20, 181-196, (1985) · Zbl 0578.60082
[7] Ross, S.M., A model in which component failure rates depend on the working set, Naval res. logist. quart., 31, 297-301, (1984) · Zbl 0538.62087
[8] Shaked, M.; Shanthikumar, J.G., Multivariate imperfect repair, Oper. res., 34, 437-448, (1986) · Zbl 0616.62129
[9] Shaked, M.; Shanthikumar, J.G., Multivariate stochastic orderings and positive dependence in reliability theory, Math. oper. res., 15, 545-552, (1990) · Zbl 0714.60078
[10] Whitt, W., Multivariate monotone likelihood ratio and uniform conditional stochastic order, J. appl. probab., 19, 695-701, (1982) · Zbl 0487.60015
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