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Moment inequalities for sums of DMRL random variables

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Enrico Fagiuoli  and
Franco Pellerey
Enrico Fagiuoli*
Affiliation:
Università di Milano
Franco Pellerey*
Affiliation:
Università di Urbino
*
∗Postal address: Dipartimento di Matematica, Università di Milano, Via L. Cicognara 7, 20129 Milano, Italy.
∗∗Postal address: Istituto di Biomatematica, Università di Urbino, Via Saffi 1, 61029 Urbino, Italy.
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Abstract

Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.

Keywords

MOMENT INEQUALITIESAGEING PROPERTIESDMRLNBUEHNBUESUMS OF INDEPENDENT VARIABLESMR STOCHASTIC ORDERMOMENT GENERATING FUNCTIONS

MSC classification

Primary: 60E05: Distributions
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 525 - 535
Copyright
Copyright © Applied Probability Trust 1997 

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References

Alzaid, A., Kim, J. S. and Proschan, F. (1991) Laplace ordering and its applications. J. Appl. Prob. 28, 116–130.Google Scholar
Barlow, R., Marshall, A. W. and Proschan, F. (1963) Properties of probability distributions with monotone hazard rate. Ann. Math. Statist. 34, 375–389.Google Scholar
Barlow, R. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With, Silver Spring, MD.Google Scholar
Bhattacharjee, M. C. (1982) The class of mean residual lives and some consequences. SIAM J. Alg. Disc. Meth. 3, 56–65.Google Scholar
Cox, D. R. (1962) Renewal Theory. Wiley, New York.Google Scholar
Di Crescenzo, A. (1995) Personal communication.Google Scholar
Fagiuoli, E. and Pellerey, F. (1993) New partial orderings and applications. Naval Res. Logist. 40, 829–842.Google Scholar
Gupta, P. L. and Gupta, R. C. (1983) On the moments of residual life in reliability and some characterization results. Commun. Statist. – Theory Meth. 12, 449–461.CrossRefGoogle Scholar
Joag-Dev, K., Kochar, S. and Proschan, F. (1995) A general composition theorem and its applications to certain partial orderings of distributions. Statist. Prob. Lett. 22, 111–119.CrossRefGoogle Scholar
Karlin, S. (1968) Total Positivity. Vol. I. Stanford University Press, Stanford, CA.Google Scholar
Klefsjö, B. (1982) The HNBUE and HNWUE classes of life distributions. Naval Res. Logist. Quart. 29, 331–344.Google Scholar
Kopocinska, I. and Kopocinsky, B. (1985) The DMRL closure problem. Bullet. Polish Acad. Sci. Math. 33, 425–429.Google Scholar
Launer, R. L. (1984) Inequalities for NBUE and NWUE life distributions. Operat. Res. 32, 660–667.Google Scholar
Massey, W. A. and Whitt, W. (1993) A probabilistic generalization of Taylor's theorem. Statist. Prob. Lett. 16, 51–54.Google Scholar
Sengupta, D. (1994) Another look at the moment bounds on reliability. J. Appl. Prob. 31, 777–787.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1994) Stochastic Orders and their Applications. Academic Press, New York.Google Scholar
Singh, H. (1989) On partial orderings of life distributions. Naval Res. Logist. 36, 103–110.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and other Stochastic Models. Wiley, New York.Google Scholar
Whitt, W. (1985) The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.Google Scholar