## The Annals of Statistics

### A Bayes method for a monotone hazard rate via S-paths

Man-Wai Ho

#### Abstract

A class of random hazard rates, which is defined as a mixture of an indicator kernel convolved with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of S-paths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over S-paths. The path characterization or the estimator is proved to be a Rao–Blackwellization of an existing partition characterization or partition-sum estimator. This accentuates the importance of S-paths in Bayesian modeling of monotone hazard rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to approximate this class of estimates. It is shown that S-path characterization also exists in modeling with covariates by a proportional hazard model, and the proposed algorithm again applies. Numerical results of the method are given to demonstrate its practicality and effectiveness.

#### Article information

Source
Ann. Statist., Volume 34, Number 2 (2006), 820-836.

Dates
First available in Project Euclid: 27 June 2006

https://projecteuclid.org/euclid.aos/1151418242

Digital Object Identifier
doi:10.1214/009053606000000047

Mathematical Reviews number (MathSciNet)
MR2283394

Zentralblatt MATH identifier
1092.62035

Subjects
Primary: 62G05: Estimation
Secondary: 62F15: Bayesian inference

#### Citation

Ho, Man-Wai. A Bayes method for a monotone hazard rate via S -paths. Ann. Statist. 34 (2006), no. 2, 820--836. doi:10.1214/009053606000000047. https://projecteuclid.org/euclid.aos/1151418242

#### References

• Aalen, O. (1975). Statistical inference for a family of counting processes. Ph.D. dissertation, Univ. California, Berkeley.
• Aalen, O. (1978). Nonparametric inference for a family of counting processes. Ann. Statist. 6 701–726.
• Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
• Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference Under Order Restrictions. The Theory and Application of Isotonic Regression. Wiley, New York.
• Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241.
• Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536–561.
• Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355.
• Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. in Appl. Probab. 31 929–953.
• Brunner, L. J. (1995). Bayesian linear regression with error terms that have symmetric unimodal densities. J. Nonparametr. Statist. 4 335–348.
• Brunner, L. J. and Lo, A. Y. (1989). Bayes methods for a symmetric unimodal density and its mode. Ann. Statist. 17 1550–1566.
• Brunner, L. J. and Lo, A. Y. (1994). Nonparametric Bayes methods for directional data. Canad. J. Statist. 22 401–412.
• Cox, D. R. (1972). Regression models and life-tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187–220.
• Drǎgichi, L. and Ramamoorthi, R. V. (2003). Consistency of Dykstra–Laud priors. Sankhyā 65 464–481.
• Dykstra, R. L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356–367.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
• Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Analysis Machine Intelligence 6 721–741.
• Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125–153.
• Hald, A. (1981). T. N. Thiele's contributions to statistics. Internat. Statist. Rev. 49 1–20.
• Hall, P., Huang, L.-S., Gifford, J. A. and Gijbels, I. (2001). Nonparametric estimation of hazard rate under the constraint of monotonicity. J. Comput. Graph. Statist. 10 592–614.
• Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
• Hayakawa, Y., Zukerman, J., Paul, S. and Vignaux, T. (2001). Bayesian nonparametric testing of constant versus nondecreasing hazard rates. In System and Bayesian Reliability: Essays in Honor of Professor Richard E. Barlow on His $70$th Birthday (Y. Hayakawa, T. Irony and M. Xie, eds.) 391–406. World Scientific, River Edge, NJ.
• Ho, M.-W. (2002). Bayesian inference for models with monotone densities and hazard rates. Ph.D. dissertation, Dept. Information and Systems Management, Hong Kong Univ. of Science and Technology.
• Ho, M.-W. (2005). A Bayes method for an asymmetric unimodal density with mode at zero. Unpublished manuscript.
• Ho, M.-W. (2006). Bayes estimation of a symmetric unimodal density via S-paths. J. Comput. Graph. Statist. To appear.
• Ho, M.-W. and Lo, A. Y. (2001). Bayesian nonparametric estimation of a monotone hazard rate. In System and Bayesian Reliability: Essays in Honor of Professor Richard E. Barlow on His $70$th Birthday (Y. Hayakawa, T. Irony and M. Xie, eds.) 301–314. World Scientific, River Edge, NJ.
• Huang, J. and Wellner, J. A. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Statist. 22 3–33.
• Ibrahim, J. G., Chen, M.-H. and MacEachern, S. N. (1999). Bayesian variable selection for proportional hazards models. Canad. J. Statist. 27 701–717.
• Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161–173.
• Ishwaran, H. and James, L. F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13 1211–1235.
• Ishwaran, H. and James, L. F. (2004). Computational methods for multiplicative intensity models using weighted gamma processes: Proportional hazards, marked point processes and panel count data. J. Amer. Statist. Assoc. 99 175–190.
• James, L. F. (2003). Bayesian calculus for gamma processes with applications to semiparametric intensity models. Sankhyā 65 179–206.
• James, L. F. (2005). Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33 1771–1799.
• James, L. F. and Lau, J. W. (2005). A class of generalized hyperbolic continuous time integrated stochastic volatility likelihood models. Available at www.arXiv.org/abs/math.ST/0503056.
• James, L. F., Lijoi, A. and Prünster, I. (2005). Bayesian inference via classes normalized random measures. Available at www.arXiv.org/abs/math.ST/0503394.
• Kalbfleisch, J. D. (1978). Non-parametric Bayesian analysis of survival time data. J. Roy. Statist. Soc. Ser. B 40 214–221.
• Kingman, J. F. C. (1967). Completely random measures. Pacific J. Math. 21 59–78.
• Kingman, J. F. C. (1993). Poisson Processes. Oxford Univ. Press, New York.
• Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Z. Wahrsch. Verw. Gebiete 59 55–66.
• Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351–357.
• Lo, A. Y., Brunner, L. J. and Chan, A. T. (1996). Weighted Chinese restaurant processes and Bayesian mixture models. Research report, Hong Kong Univ. of Science and Technology. Available at www.erin.utoronto.ca/~jbrunner/papers/wcr96.pdf.
• Lo, A. Y. and Weng, C.-S. (1989). On a class of Bayesian nonparametric estimates. II. Hazard rate estimates. Ann. Inst. Statist. Math. 41 227–245.
• Lo, S. H. and Phadia, E. (1992). On estimation of a survival function in reliability theory based on censored data. Preprint, Columbia Univ.
• McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.
• Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.
• Mykytyn, S. W. and Santner, T. J. (1981). Maximum likelihood estimation of the survival function based on censored data under hazard rate assumptions. Comm. Statist. A–-Theory Methods 10 1369–1387.
• Padgett, W. J. and Wei, L. J. (1980). Maximum likelihood estimation of a distribution function with increasing failure rate based on censored observations. Biometrika 67 470–474.
• Prakasa Rao, B. L. S. (1970). Estimation of distributions with monotone failure rate. Ann. Math. Statist. 41 507–519.
• Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701–1762.
• Villalobos, M. and Wahba, G. (1987). Inequality-constrained multivariate smoothing splines with application to the estimation of posterior probabilities. J. Amer. Statist. Assoc. 82 239–248.
• Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85 251–267.