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Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates

Part of: Survival analysis and censored data

Published online by Cambridge University Press:  14 July 2016

Henry W. Block ,
Naftali A. Langberg ,
Thomas H. Savits  and
Jie Wang
Henry W. Block*
Affiliation:
University of Pittsburgh
Naftali A. Langberg*
Affiliation:
University of Haifa
Thomas H. Savits*
Affiliation:
University of Pittsburgh
Jie Wang*
Affiliation:
University of Pittsburgh
*
∗Postal address: Department of Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
∗∗∗∗Postal address: Department of Statistics, University of Haifa, Mount Carmel, Haifa 31999, Israel.
∗Postal address: Department of Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
∗Postal address: Department of Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
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Abstract

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It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

Keywords

Reliabilityfailure rate functionbathtub failure rate

MSC classification

Secondary: 62N05: Reliability and life testing
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 899 - 907
Copyright
Copyright © Applied Probability Trust 2010 

References

Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behaviour of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721–740.CrossRefGoogle Scholar
Block, H. W., Li, Y., Savits, T. H. and Wang, J. (2008). Continuous mixtures with bathtub-shaped failure rates. J. Appl. Prob. 45, 260–270.CrossRefGoogle Scholar
Glaser, R. E. (1980). Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75, 667–672.CrossRefGoogle Scholar
Gupta, R. C. and Warren, R. (2001). Determination of change points of non-monotonic failure rates. Commun. Statist. Theory Meth. 30, 1903–1903.CrossRefGoogle Scholar
Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.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
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar