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On Bayes inference for a bathtub failure rate via S-paths

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Abstract

A class of semi-parametric hazard/failure rates with a bathtub shape is of interest. It does not only provide a great deal of flexibility over existing parametric methods in the modeling aspect but also results in a closed-form and tractable Bayes estimator for the bathtub-shaped failure rate. Such an estimator is derived to be a finite sum over two S-paths due to an explicit posterior analysis in terms of two (conditionally independent) S-paths. These, newly discovered, explicit results can be proved to be Rao–Blackwell improvements of counterpart results in terms of partitions that are readily available by a specialization of James’ work (Ann Stat 33:1771–1799, 2005). Both iterative and non-iterative computational procedures are introduced for evaluating the hazard estimates. Two applications of the proposed methodology are discussed, of which one is about a Bayesian test for bathtub-shaped failure rates and the other is related to modeling with covariates.

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References

  • Aalen, O. (1975). Statistical inference for a family of counting processes. Ph.D. dissertation. University of California, Berkeley.

  • Aalen O. (1978) Nonparametric inference for a family of counting processes. The Annals of Statistics 6: 701–726

    Article  MathSciNet  MATH  Google Scholar 

  • Aarset M.V. (1985) The null distribution of a test of constant versus bathtub-failure rate. Scandinavian Journal of Statistics 12: 55–62

    MathSciNet  MATH  Google Scholar 

  • Amman L. (1984) Bayesian nonparametric inference for quantal response data. The Annals of Statistics 12: 636–645

    Article  MathSciNet  Google Scholar 

  • Barlow R.E., Bartholomew D.J., Bremner J.M., Brunk H.D. (1972) Statistical inference under order restrictions. Wiley, New York

    MATH  Google Scholar 

  • Barron A., Schervish M.J., Wasserman L. (1999) The consistency of posterior distributions in nonparametric problems. The Annals of Statistics 27: 536–561

    Article  MathSciNet  MATH  Google Scholar 

  • Bergman B. (1979) On age replacement and the total time on test concept. Scandinavian Journal of Statistics 6: 161–168

    MathSciNet  MATH  Google Scholar 

  • Bickel P.J., Doksum K.A. (1969) Tests for monotone failure rate based on normalized spacings. The Annals of Mathematical Statistics 40: 1216–1235

    Article  MathSciNet  MATH  Google Scholar 

  • Bray, T., Crawford, G., Proschan, F. (1967a). Maximum likelihood estimation of a failure rate function initially increasing and then decreasing, In Proceedings 1967 Spring Seminar on Reliability, Maintainability, IEEE.

  • Bray, T., Crawford, G., Proschan, F. (1967b). Maximum likelihood estimation of a U-shaped failure rate function. Mathematical Note 534. Mathematics Research Laboratory, Boeing Scientific Research Laboratories, Seattle, WA. http://www.stat.washington.edu/jaw/RESEARCH/OLD-PAPERS-OTHERS/UMLE.pdf

  • Brunner L.J. (1992) Bayesian nonparametric methods for data from a unimodal density. Statistics and Probability Letters 14: 195–199

    Article  MathSciNet  MATH  Google Scholar 

  • Brunner L.J., Lo A.Y. (1989) Bayes methods for a symmetric unimodal density and its mode. The Annals of Statistics 17: 1550–1566

    Article  MathSciNet  MATH  Google Scholar 

  • Cox D.R. (1972) Regression models and life-tables (with discussion). Journal of the Royal Statistical Society. Series B. Statistical Methodology 34: 187–220

    MATH  Google Scholar 

  • Drǎgichi L., Ramamoorthi R.V. (2003) Consistency of Dykstra–Laud priors. Sankhyā 65: 464–481

    Google Scholar 

  • Dykstra R.L., Laud P. (1981) A Bayesian nonparametric approach to reliability. The Annals of Statistics 9: 356–367

    Article  MathSciNet  MATH  Google Scholar 

  • Gail M.H., Gastwirth J.L. (1978a) A scale-free goodness-of-fit test for the exponential distribution based on the Gini statistic. Journal of the Royal Statistical Society Series B, Statistical Methodology 40: 350–357

    MathSciNet  MATH  Google Scholar 

  • Gail M.H., Gastwirth J.L. (1978b) A scale-free goodness-of-fit test for the exponential distribution based on the Lorenz curve. Journal of the American Statistical Association 73: 787–793

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh J.K., Ramamoorthi R.V. (2003) Bayesian nonparametrics. Springer, New York

    MATH  Google Scholar 

  • Gijbels I., Heckman N. (2004) Nonparametric testing for a monotone hazard function via normalized spacings. Journal of Nonparametric Statistics 16: 463–478

    Article  MathSciNet  MATH  Google Scholar 

  • Hald A. (1981) T. N. Thieles contributions to statistics. International Statistical Review 49: 1–20

    Article  MathSciNet  MATH  Google Scholar 

  • Hall P., Van Keilegom I. (2005) Testing for monotone increasing hazard rate. The Annals of Statistics 33: 1109–1137

    Article  MathSciNet  MATH  Google Scholar 

  • Haupt E., Schabe H. (1997) The TTT transformation and a new bathtub distribution model. Journal of Statistical Planning and Inference 60: 229–240

    Article  MathSciNet  MATH  Google Scholar 

  • Ho M.-W. (2006a) A Bayes method for a monotone hazard rate via S-paths. The Annals of Statistics 34: 820–836

    Article  MathSciNet  MATH  Google Scholar 

  • Ho M.-W. (2006b) Bayes estimation of a symmetric unimodal density via S-paths. Journal of Computational and Graphical Statistics 15: 848–860

    Article  MathSciNet  Google Scholar 

  • Ibrahim J.G., Chen M.-H., MacEachern S.N. (1999) Bayesian variable selection for proportional hazards models. The Canadian Journal of Statistics 37: 701–717

    Article  MathSciNet  Google Scholar 

  • Ishwaran H., James L.F. (2003) Generalized weighted Chinese restaurant processes for species sampling mixture models. Statistica Sinica 13: 1211–1235

    MathSciNet  MATH  Google Scholar 

  • Ishwaran H., James L.F. (2004) Computational methods for multiplicative intensity models using weighted gamma processes: proportional Hazards, marked point processes, and panel count data. Journal of the American Statistical Association 99: 175–190

    Article  MathSciNet  MATH  Google Scholar 

  • James L.F. (2003) Bayesian calculus for gamma processes with applications to semiparametric intensity models. Sankhyā 65: 196–223

    MATH  Google Scholar 

  • James L.F. (2005) Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. The Annals of Statistics 33: 1771–1799

    Article  MathSciNet  MATH  Google Scholar 

  • Jankowski, H., Wellner, J. A. (2008a) Nonparametric estimation of a convex bathtub-shaped hazard function. Technical Report 521, Department of Statistics, University of Washington. http://arXiv:0801.0712v1[math.ST].

  • Jankowski, H., Wellner, J. A. (2008b) Computation of Nonparametric Convex Hazard Estimators via Profile Methods. Technical Report 542, Department of Statistics, University of Washington.

  • Kalbfleisch J.D. (1978) Non-parametric Bayesian analysis of survival time data. Journal of the Royal Statistical Society Series B, Statistical Methodology 40: 214–221

    MathSciNet  MATH  Google Scholar 

  • Kalbfleisch J.D., Prentice R.L. (1980) The statistical analysis of failure time data. Wiley, New York

    MATH  Google Scholar 

  • Kass R.E., Raftery A.E. (1995) Bayes factors and model uncertainty. Journal of the American Statistical Associaton 90: 773–795

    Article  MATH  Google Scholar 

  • Kingman J.F.C. (1967) Completely random measures. Pacific Journal of Mathematics 21: 59–78

    MathSciNet  MATH  Google Scholar 

  • Kingman J.F.C. (1993) Poisson processes. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Kong A., Liu J.S., Wong W.H. (1994) Sequential imputations and Bayesian missing data problems. Journal of the American Statistical Associaton 89: 278–288

    Article  MATH  Google Scholar 

  • Lai C.D., Xie M., Murthy D.N.P. (2001) Bathtub-shaped failure rate distributions. In: Balakrishnan N., Rao C.R. (eds) Handbook of statistics 20. Elsevier, London, pp 69–104

    Google Scholar 

  • Laud P., Damien P., Walker S.G. (2006) Computations via auxiliary random functions for survival models. Scandinavian Journal of Statistics 33: 219–226

    Article  MathSciNet  MATH  Google Scholar 

  • Lawless J.F. (1982) Statistical methods and methods for life time data. Wiley, New York

    Google Scholar 

  • Liu J., Chen R. (1998) Sequential Monte Carlo methods for dynamic systems. Journal of the American Statistical Associaton 93: 1032–1044

    Article  MATH  Google Scholar 

  • Lo A.Y., Brunner L.J., Chan A.T. (1996). Weighted Chinese restaurant processes and Bayesian mixture models, Research report 1, The Hong Kong University of Science and Technology, Department of Information and Systems Management. http://www.erin.utoronto.ca/~jbrunner/papers/wcr96.pdf

  • McCullagh P. (1987) Tensor methods in statistics. Chapman & Hall, London

    MATH  Google Scholar 

  • Meyn S.P., Tweedie R.L. (1993) Markov chains and stochastic stability. Springer, Berlin

    MATH  Google Scholar 

  • Mitra M., Basu S.K. (1995) Change point estimation in non-monotonic aging models. Annals of the Institute of Statistical Mathematics 47: 483–491

    MathSciNet  MATH  Google Scholar 

  • Proschan F., Pyke R. (1967) Tests for monotone failure rate. In Fifth Berkeley Symposium 3: 293–313

    MathSciNet  Google Scholar 

  • Rajarshi S., Rajarshi M.B. (1988) Bathtub distributions: a review. Communications in Statistics. Theory and Methods 17: 2597–2621

    Article  MathSciNet  MATH  Google Scholar 

  • Reboul L. (2005) Estimation of a function under shape restrictions. Applications to reliability. The Annals of Statistics 33: 1330–1356

    MathSciNet  MATH  Google Scholar 

  • Silverman B.W. (1981) Using kernel density estimates to investigate multimodality. Journal of the Royal Statistical Society. Series B. Statistical Methodology 43: 97–99

    Google Scholar 

  • Singpurwalla N.D. (2006) Reliability and risk: a Bayesian perspective. Wiley, Chichester

    MATH  Google Scholar 

  • Vaurio J.K. (1999) Indentification of process and distribution characteristics by testing monotonic and non-monotonic trends in failure intensities and hazard rates. Reliability Engineering and System Safety 64: 345–357

    Article  Google Scholar 

Download references

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  1. Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore, 117546, Singapore

    Man-Wai Ho

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Ho, MW. On Bayes inference for a bathtub failure rate via S-paths. Ann Inst Stat Math 63, 827–850 (2011). https://doi.org/10.1007/s10463-009-0253-1

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  • DOI: https://doi.org/10.1007/s10463-009-0253-1

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