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Mixtures of distributions with increasing linear failure rates

Part of: Special processes Survival analysis and censored data

Published online by Cambridge University Press:  14 July 2016

Henry W. Block ,
Thomas H. Savits  and
Eshetu T. Wondmagegnehu
Henry W. Block*
Affiliation:
University of Pittsburgh
Thomas H. Savits*
Affiliation:
University of Pittsburgh
Eshetu T. Wondmagegnehu*
Affiliation:
Western Michigan University
*
∗ Postal address: Department of Statistics, University of Pittsburgh, PA 15260, USA.
∗ Postal address: Department of Statistics, University of Pittsburgh, PA 15260, USA.
∗∗∗ Postal address: Department of Statistics, Western Michigan University, Kalamazoo, MI 49008, USA.
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Abstract

Populations of specific components are often heterogeneous and consist of a small number of different sub-populations. For example there are often two groups: defective components with shorter lifetimes and standard components with longer lifetimes. Another heterogeneous population results when components produced by two different manufacturing lines are combined. In either case a mixture results. The resulting population can be described using the statistical concept of a mixture. It is a well-known result that a mixture of distributions with decreasing failure rates has a decreasing failure rate. However, little is known about the monotonicity of a mixture when the various subpopulations have failure rates which are not necessarily decreasing. In this paper we study and attempt to determine the shape as well as the overall behavior of the failure rate of a mixture from two subpopulations each of which has increasing linear failure rate.

Keywords

Mixtureincreasing linear failure rateincreasing failure ratebathtub failure ratemodified bathtub shape

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62N05: Reliability and life testing
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 485 - 504
Copyright
Copyright © Applied Probability Trust 2003 

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References

Block, H. W., and Joe, H. (1997). The behavior of the failure rate functions of mixture. Lifetime Data Anal. 3, 269–288.CrossRefGoogle Scholar
Block, H. W., and Savits, T. H. (1997). Burn-in. Statist. Sci. 12, 1–19.CrossRefGoogle Scholar
Block, H. W., and Savits, T. H. (2001). Comments on ‘Understanding the shape of the hazard rate: a process point of view’ by O. O. Aalen and H. Gjessing. Statist. Sci. 16, 14–16.Google Scholar
Block, H. W., Li, Y., and Savits, T. H. (2003). Initial and final behavior of failure rate functions for mixtures and systems. To appear in J. Appl. Prob. 40, No. 3.CrossRefGoogle Scholar
Block, H. W., Mi, J., and Savits, T. H. (1993). Burn-in and mixed populations. J. Appl. Prob. 30, 692–702.CrossRefGoogle Scholar
Clarotti, C. A., and Spizzichino, F. (1990). Bayes burn-in decision procedure. Prob. Eng. Inf. Sci. 4, 437–445.CrossRefGoogle Scholar
Glaser, R. E. (1980). Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75, 667–672.CrossRefGoogle Scholar
Gupta, P. L., and Gupta, R. C. (1996). Ageing characteristics of the Weibull mixtures. Prob. Eng. Inf. Sci. 10, 591–600.CrossRefGoogle Scholar
Gurland, J., and Sethuraman, J. (1994). Reversal of increasing failure rate when pooling failure data. Technometrics 36, 416–418.Google Scholar
Lynch, J. D. (1999). On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate. Prob. Eng. Inf. Sci. 13, 33–36.CrossRefGoogle Scholar
Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics 5, 375–383.CrossRefGoogle Scholar
Rajarshi, S., and Rajarshi, M. B. (1988). Bathtub distributions: a review. Commun. Statist. Theory Meth. 17, 2597–2621.CrossRefGoogle Scholar