Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T15:35:41.123Z Has data issue: false hasContentIssue false
  • English
  • Français

Another look at the moment bounds on reliability

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Debasis Sengupta
Debasis Sengupta*
Affiliation:
Indian Statistical Institute, Calcutta
*
∗ Postal address: Applied Statistics, Surveys and Computing Division, Indian Statistical Institute, 203 Barrackpore Trunk Road, Calcutta 700 035, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper a unified derivation of the upper and lower bounds (in terms of the mean) of an IFR, DFR, IFRA, DFRA, NBU or NWU reliability function is presented. The method of proof provides a simpler alternative to the various proofs known in the literature. Moreover, this method can be used to generalize the existing results in two ways, as demonstrated here. First, the bounds for the reliability function are obtained in terms of any finite moment in all these cases. Subsequently we provide a moment bound on a reliability function which ages faster or slower than a known one in some sense. The existing literature does not offer any of these generalizations, except for a few results which are available in an unnecessarily complicated form.

Keywords

CONVEXITYSTAR-SHAPEDNESSSUPERADDITIVITYRELIABILITY BOUNDSHAZARD FUNCTIONIFRDFRIFRADFRANBUNWU

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 777 - 787
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Holt Reinhart and Winston, New York.Google Scholar
Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt Reinhart and Winston, New York.Google Scholar
Haines, A. and Singpurwala, N. D. (1974) Some contributions to the stochastic characterization of wear. In Reliability and Biometry, ed. Proschan, F. and Serfling, R. J., pp. 47–80. SIAM, Philadelphia.Google Scholar
Kalashnikov, V. V. and Rachev, S. T. (1986) Characterization of queuing models and their stability. In Probability Theory and Mathematical Statistics, ed. Prohorov, Yu. K. et al. 2, pp. 37–53. VNU Science Press.Google Scholar
Klefsjö, B. (1982) The HNBUE and HNWUE classes of life distributions. Naval Res. Logist. Quart. 29, 331–344.Google Scholar
Korzeniowski, A. and Opawski, A. (1976) Bounds for reliability in the NBU, NWU, NBUE and NWUE classes. Zast. Mat. Appl. Math. 15, 1–5.Google Scholar
Marshall, A. W. and Proschan, F. (1972) Classes of distributions applicable in replacement with renewal theory implications. Proc. 6th Berkeley Sym. Math. Statist. Prob. 395–415.Google Scholar