# The Failure Rates of Mixtures

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## Abstract

Mixtures of distributions of lifetimes occur in many settings. In engineering applications, it is often the case that populations are heterogeneous, often with a small number of subpopulations. In survival analysis, selection effects can often occur. The concept of a failure rate in these settings becomes a complicated topic, especially when one attempts to interpret the shape as a function of time. Even if the failure rates of the subpopulations of the mixture have simple geometric or parametric forms, the shape of the mixture is often not transparent.

Recent results, developed by the author (with Joe, Li, Mi, Savits, and Wondmagegnehu) in a series of papers, are presented. These results focus on general results concerning the asymptotic limit and eventual monotonicity of a mixture, and also the overall behavior for mixtures of specific parametric families.

An overall picture is given of different things that influence the behavior of the failure rate of a mixture.

## Keywords and phrases

Failure rate mixture coherent systems signature## Preview

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## References

- 1.Barlow, R.E., and Proschan, F. (1975).
*Statistical Theory of Reliability*, Holt, Rinehart and Winston, New York.zbMATHGoogle Scholar - 2.Berg, H., Bienvenu, M., and Cleroux, R. (1986). Age replacement policy with age-dependent minimal repair,
*Informatics*,**24**, 26–32.zbMATHGoogle Scholar - 3.Block, H. W., Dugas, M., and Samaniego, F. J. (2004). Signature-related results on failure rates and lifetimes, submitted for publication.Google Scholar
- 4.Block, H., and Joe, H. (1997). Tail behavior of the failure rate functions of mixtures,
*Lifetime Data Analysis***3**, 269–288.zbMATHCrossRefGoogle Scholar - 5.Block, H. W., Li, Y., and Savits, T. H. (2003a). Preservation of properties under mixtures,
*Probability in Engineering and Information Sciences*,**17**, 205–212.zbMATHCrossRefMathSciNetGoogle Scholar - 6.Block, H. W., Li, Y. and Savits, T. H. (2003b). Initial and final behavior of failure rate functions for mixtures and systems,
*Journal of Applied Probability*,**40**, 721–740.zbMATHCrossRefMathSciNetGoogle Scholar - 7.Block, H. W., Li, Y., and Savits, T. H. (2004). Mixtures of normal distributions: modality and failure rate,
*Technical Report*, University of Pittsburgh, Pittsburgh, PA.Google Scholar - 8.Block, H. W., Mi, J., and Savits, T. H. (1993). Burn-in and mixed populations,
*Journal of Applied Probability*,**30**, 692–702.zbMATHCrossRefMathSciNetGoogle Scholar - 9.Block, H. W., and Savits, T. H. (1997). Burn-in,
*Statistical Science*,**12**, 1–19.CrossRefGoogle Scholar - 10.Block, H. W., Savits, T. H., and Singh, H. (2002). A criterion for burnin which balances mean residual life and residual variance,
*Operations Research*,**50**, 290–296.CrossRefMathSciNetzbMATHGoogle Scholar - 11.Block, H. W., Savits, T. H., and Wondmagegnehu, E. (2003). Mixtures of distributions with linear failure rates,
*Journal of Applied Probability*,**40**, 485–504.zbMATHCrossRefMathSciNetGoogle Scholar - 12.Bretagnolle, J., and Huber-Carol, C. (1988). Effects of omitting covariates in Cox’s model for survival data,
*Scandinavian Journal of Statistics*,**15**, 125–138.MathSciNetzbMATHGoogle Scholar - 13.Chen, C. S., and Savits, T. H. (1992). Optimal age and block replacement for a general maintenance model,
*Probability in Engineering and Information Sciences*,**6**, 81–98.zbMATHCrossRefGoogle Scholar - 14.Glaser, R. E. (1980). Bathtub and related failure rate characterizations,
*Journal of the American Statistical Association*,**75**, 667–672.zbMATHCrossRefMathSciNetGoogle Scholar - 15.Gupta, R. C., and Warren, R. (2001). Determination of change points of non-monotonic failure rates,
*Communications in Statistics—Theory and Methods*,**30**, 1903–1920.zbMATHCrossRefMathSciNetGoogle Scholar - 16.Gurland, J., and Sethuraman, J. (1994). Reversal of increasing failure rates when pooling failure data,
*Technometrics*,**36**, 416–418.zbMATHCrossRefGoogle Scholar - 17.Gurland, J., and Sethuraman, J. (1995). How pooling failure data may reverse increasing failure rates,
*Journal of the American Statistical Association*,**90**, 1416–1423.zbMATHCrossRefMathSciNetGoogle Scholar - 18.Jensen, F., and Petersen, N. E. (1982).
*Burn-in*, John Wiley_& Sons, New York.Google Scholar - 19.Jiang, R., and Murthy, D. N. P. (1998). Mixture of Weibull distributions: parametric characterization of failure rate functions,
*Applied Stochastic Models in Data Analysis*,**14**, 47–65.zbMATHCrossRefMathSciNetGoogle Scholar - 20.Klein, J. P., and Moeschberger, M. L. (1997).
*Survival Analysis*, Springer-Verlag, New York.zbMATHGoogle Scholar - 21.Lynch, J. D. (1999). On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate,
*Probability in Engineering and Information Sciences*,**13**, 33–36.zbMATHCrossRefMathSciNetGoogle Scholar - 22.Mi, J. (1996). Minimizing some cost functions related to both burn-in and field use,
*Operations Research***44**, 497–500.zbMATHGoogle Scholar - 23.Navarro, J., and Hernandez, P. J. (2002). How to obtain bathtub-shaped failure rate models from normal mixtures
*Probability in Engineering and Information Sciences*,**18**, 511–531.MathSciNetGoogle Scholar - 24.Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate,
*Technometrics*,**5**, 373–383.CrossRefGoogle Scholar - 25.Puri, P. S., and Singh, H. (1986). Optimum replacement of a system subject to shocks: a mathematical lemma,
*Operations Research*,**34**, 782–789.zbMATHMathSciNetCrossRefGoogle Scholar - 26.Robertson, C. A., and Fryer, J. G. (1969). Some descriptive properties of normal mixtures,
*Skandinavisk Aktuarietidskrift*,**52**, 137–146.MathSciNetGoogle Scholar - 27.Samaniego, F. (1985). On the closure of the IFR class under the formation of coherent systems,
*IEEE Transactions on Reliability*,**R-34**, 69–72.CrossRefGoogle Scholar - 28.Savits, T. H. (2003). Preservation of generalized bathtub functions,
*Journal of Applied Probability*,**40**, 1–12.CrossRefMathSciNetGoogle Scholar - 29.Schilling, M. F., Watkins, A. E., and Watkins, W. (2002). Is human height bimodal?
*The American Statistician*,**56**, 223–229.CrossRefMathSciNetGoogle Scholar - 30.Vaupel, T. W., and Yashin, A. I. (1985). Heterogeneity’s ruse: some surprising effects of selection on population dynamics,
*The American Statistician*,**39**, 176–185.CrossRefMathSciNetGoogle Scholar - 31.Wang, J.-L., Muller, H., and Capra, W. B. (1998). Analysis of oldest-old mortality,
*The Annals of Statistics*,**26**, 126–133.zbMATHCrossRefMathSciNetGoogle Scholar - 32.Wondmagegnehu, E. T. (2002). Mixture of distributions with increasing failure rates, Ph.D. Thesis, University of Pittsburgh, Pittsburgh, PA.Google Scholar
- 33.Wondmagegnehu, E. T., Navarro, J., and Hernandez, P. J. (2005). Bathtub shaped failure rates from mixtures: A practical point of view,
*IEEE Transactions on Reliability*,**54**, 270–275.CrossRefGoogle Scholar