Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T12:31:21.640Z Has data issue: false hasContentIssue false
  • English
  • Français

TP2-Orderings and the IFR Property with Applications

Published online by Cambridge University Press:  27 July 2009

S. Durham ,
J. Lynch  and
W.J. Padgett
S. Durham
Affiliation:
Department of Statistics University of South Carolina, Columbia, South Carolina 29208
J. Lynch
Affiliation:
Department of Statistics University of South Carolina, Columbia, South Carolina 29208
W.J. Padgett
Affiliation:
Department of Statistics University of South Carolina, Columbia, South Carolina 29208
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper, TP2 orderings of distributions and of survival functions are considered. It is shown that the first passage time of a Markov process with TP2- ordered transition distributions has an increasing failure rate (IFR). Conditions are also given for which mixtures of IFR distributions are IFR. A formula is obtained for the failure rate when the strength distribution is a function of both load and strength. This formula, in conjunction with the TP2-ordering on the strength survival functions and log-concavity of the strength and load variables, leads to an increasing failure rate for the strength distribution. Finally, two models based on this theory are presented which explain the IFR character of the strength distribution of fibrous composite materials.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 4 , Issue 1 , January 1990 , pp. 73 - 88
Copyright
Copyright © Cambridge University Press 1990

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

Assaf, D., Shaked, M., & Shanthikumar, J.G. (1985). First passage times with PFr densities. Journal of Applied Probability 22: 185–196.CrossRefGoogle Scholar
Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing: Probability models. New York:Holt, Rinehart and Winston, Inc.Google Scholar
Brown, M. & Chaganty, N.R. (1983). On the first passage time distribution for a class of Markov chains. Annals of Probability 11: 1000–1008.CrossRefGoogle Scholar
Daniels, H.E. (1945). The statistical theory of the strength of a bundle of threads, I: Proceedings of the Royal Society A 183: 404–435.Google Scholar
Durham, S., Lynch, J., & Padgett, W.J. (1988). Inference for strength distributions of brittle fibers under increasing failure rate. Journal of Composite Materials 22: 1131–1140.CrossRefGoogle Scholar
Karlin, S. (1964). Total positivity, absorption probabilities, and applications. Transactions of the American Mathematical Society 111: 33–107.CrossRefGoogle Scholar
Karlin, S. (1968). Total positivity. Stanford, California: Stanford University Press.Google Scholar
Karlin, S. & Proschan, F. (1960). Polya type distributions and convolutions. Annals of Mathematical Statistics 31: 721–736.CrossRefGoogle Scholar
Keilson, J. & Sumita, U. (1982). Uniform stochastic orderings and related inequalities. Canadian Journal of Statistics 10: 181–198.CrossRefGoogle Scholar
Lynch, J., Mimmack, G., & Proschan, F. (1987). Uniform stochastic orderings and total positivity. Canadian Journal of Statistics 15: 63–69.CrossRefGoogle Scholar
Rosen, B.W. (1964). Tensile failure of fibrous composites. AIAA Journal 2: 1985–1991.CrossRefGoogle Scholar
Rosen, B.W. (1965). Mechanics of composite strengthening. Fiber Composite Materials. Metal Park, Ohio: American Society of Metals.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1988). On the first passage time of pure jump processes. Journal of Applied Probability 25: 501–509.CrossRefGoogle Scholar
Shanthikumar, J.G. (1988). DFR property of first passage times and its preservation under geometric compounding. Annals of Probability 16: 397–406.CrossRefGoogle Scholar
Taylor, H.M. (1987). A model for the failure process of semicrystalline polymer materials under fatigue. Probability in Engineering and Informational Sciences 1: 133–162.CrossRefGoogle Scholar
Vinogradov, O.P. (1976). On summing a random number of random variables having an increasing failure rate or a strongly unimodal distribution. Theory of Probability and Applications 21: 205–209.CrossRefGoogle Scholar