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Continuous Mixtures with Bathtub-Shaped Failure Rates

Part of: Operations research and management science Survival analysis and censored data

Published online by Cambridge University Press:  14 July 2016

Henry W. Block ,
Yulin Li ,
Thomas H. Savits  and
Jie Wang
Henry W. Block*
Affiliation:
University of Pittsburgh
Yulin Li*
Affiliation:
The University of Toledo
Thomas H. Savits*
Affiliation:
University of Pittsburgh
Jie Wang*
Affiliation:
University of Pittsburgh
*
∗Postal address: Department of Statistics, University of Pittsburgh, 2717 Cathedral of Learning, Pittsburgh, PA 15260, USA.
∗∗∗∗Postal address: Department of Statistics, University of Pittsburgh, 2717 Cathedral of Learning, Pittsburgh, PA 15260, USA.
∗Postal address: Department of Statistics, University of Pittsburgh, 2717 Cathedral of Learning, Pittsburgh, PA 15260, USA.
∗Postal address: Department of Statistics, University of Pittsburgh, 2717 Cathedral of Learning, Pittsburgh, PA 15260, USA.
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Abstract

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The failure rate of a mixture of even the most standard distributions used in reliability can have a complicated shape. However, failure rates of mixtures of two carefully selected distributions will have the well-known bathtub shape. Here we show that mixtures of whole families of distribtions can have a bathtub-shaped failure rate.

Keywords

Reliabilitybathtub-shaped failure rate

MSC classification

Secondary: 62N05: Reliability and life testing 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 260 - 270
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Block, H. and Joe, H. (1997). Tail behavior of the failure rate functions of mixtures. Lifetime Data Anal. 3, 269–288.Google Scholar
[2] Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behavior of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721–740.Google Scholar
[3] Block, H. W., Mi, J. and Savits, T. H. (1993). Burn-in and mixed populations. J. Appl. Prob. 30, 692–702.CrossRefGoogle Scholar
[4] Block, H. W., Savits, T. H. and Wondmagegnehu, E. (2003). Mixtures of distributions with linear failure rates. J. Appl. Prob. 40, 485–504.Google Scholar
[5] Clarotti, C.A. and Spizzichino, F. (1990). Bayes burn-in decision procedure. Prob. Eng. Inf. Sci. 4, 437–445.Google Scholar
[6] Constantine, G. M. and Savits, T. H. (1966). A multivariate Faà di Bruno formula with applications. Trans. Amer. Math. Soc. 348, 503–520.Google Scholar
[7] Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol II. John Wiley, New York.Google Scholar
[8] Finkelstein, M. S. and Esaulova, V. (2001). Why the mixture failure rate decreases. Reliab. Eng. System Safety 71, 173–177.Google Scholar
[9] Glaser, R. E. (1980). Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75, 667–672.Google Scholar
[10] Gleser, L. J. (1989). The gamma distribution as a mixture of exponential distributions. Amer. Statist. 43, 115–117.Google Scholar
[11] 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
[12] Gurland, J. and Sethuraman, J. (1994). Reversal of increasing failure rates when pooling failure data. Technometrics 36, 416–418.Google Scholar
[13] Gurland, J. and Sethuraman, J. (1995). How pooling failure data may reverse increasing failure rates. J. Amer. Statist. Assoc. 90, 1416–1423.Google Scholar
[14] Jewell, N. P. (1982). Mixtures of exponential distributions. Ann. Statist. 10, 479–484.Google Scholar
[15] Jiang, R. and Murthy, D. N. P. (1998). Mixture of Weibull distributions—parametric characterization of failure rate functions. Appl. Stoch. Models Data Anal. 14, 47–65.Google Scholar
[16] Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
[17] Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
[18] Meyer, P. A. (1966). Probability and Potentials. Blaisdell, Waltham, MA.Google Scholar
[19] Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics 5, 373–383.CrossRefGoogle Scholar
[20] Wang, J.-L., Müller, H. and Capra, W. B. (1998). Analysis of oldest-old mortality: lifetables revisited. Ann. Statist. 26, 126–163.CrossRefGoogle Scholar
[21] Wondmagegnehu, E. T., Navarro, J. and Hernandez, P. J. (2005). Bathtub shaped failure rate from mixtures: a practical point of view. IEEE Trans. Reliab. 54, 270–275.Google Scholar