×

zbMATH — the first resource for mathematics

A Bayes method for a monotone hazard rate via \(S\)-paths. (English) Zbl 1092.62035
Summary: 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 of 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 characterizations also exist 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.

MSC:
62F15 Bayesian inference
62N02 Estimation in survival analysis and censored data
62G05 Nonparametric estimation
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aalen, O. (1975). Statistical inference for a family of counting processes. Ph.D. dissertation, Univ. California, Berkeley. · Zbl 0389.62025
[2] Aalen, O. (1978). Nonparametric inference for a family of counting processes. Ann. Statist. 6 701–726. · Zbl 0389.62025
[3] Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes . Springer, New York. · Zbl 0769.62061
[4] 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. · Zbl 0246.62038
[5] 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. JSTOR: · Zbl 0983.60028
[6] Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536–561. · Zbl 0980.62039
[7] Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355. · Zbl 0276.62010
[8] Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. in Appl. Probab. 31 929–953. · Zbl 0957.60055
[9] Brunner, L. J. (1995). Bayesian linear regression with error terms that have symmetric unimodal densities. J. Nonparametr. Statist. 4 335–348. · Zbl 1380.62128
[10] Brunner, L. J. and Lo, A. Y. (1989). Bayes methods for a symmetric unimodal density and its mode. Ann. Statist. 17 1550–1566. · Zbl 0697.62003
[11] Brunner, L. J. and Lo, A. Y. (1994). Nonparametric Bayes methods for directional data. Canad. J. Statist. 22 401–412. JSTOR: · Zbl 0809.62046
[12] Cox, D. R. (1972). Regression models and life-tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187–220. JSTOR: · Zbl 0243.62041
[13] Drǎgichi, L. and Ramamoorthi, R. V. (2003). Consistency of Dykstra–Laud priors. Sankhyā 65 464–481. · Zbl 1193.62179
[14] Dykstra, R. L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356–367. · Zbl 0469.62077
[15] Feller, W. (1968). An Introduction to Probability Theory and Its Applications 1 , 3rd ed. Wiley, New York. · Zbl 0155.23101
[16] 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. · Zbl 0573.62030
[17] Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125–153. · Zbl 0077.33715
[18] Hald, A. (1981). T. N. Thiele’s contributions to statistics. Internat. Statist. Rev. 49 1–20. JSTOR: · Zbl 0467.62003
[19] 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. JSTOR: · Zbl 04568641
[20] Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109. · Zbl 0219.65008
[21] 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.
[22] 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.
[23] Ho, M.-W. (2005). A Bayes method for an asymmetric unimodal density with mode at zero. Unpublished manuscript.
[24] Ho, M.-W. (2006). Bayes estimation of a symmetric unimodal density via S -paths. J. Comput. Graph. Statist.
[25] 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.
[26] Huang, J. and Wellner, J. A. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Statist. 22 3–33. · Zbl 0827.62032
[27] Ibrahim, J. G., Chen, M.-H. and MacEachern, S. N. (1999). Bayesian variable selection for proportional hazards models. Canad. J. Statist. 27 701–717. JSTOR: · Zbl 0957.62018
[28] Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161–173. JSTOR: · Zbl 1014.62006
[29] Ishwaran, H. and James, L. F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13 1211–1235. · Zbl 1086.62036
[30] 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. · Zbl 1089.62520
[31] James, L. F. (2003). Bayesian calculus for gamma processes with applications to semiparametric intensity models. Sankhyā 65 179–206. · Zbl 1192.62081
[32] James, L. F. (2005). Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33 1771–1799. · Zbl 1078.62106
[33] 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.
[34] 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.
[35] Kalbfleisch, J. D. (1978). Non-parametric Bayesian analysis of survival time data. J. Roy. Statist. Soc. Ser. B 40 214–221. JSTOR: · Zbl 0387.62030
[36] Kingman, J. F. C. (1967). Completely random measures. Pacific J. Math. 21 59–78. · Zbl 0155.23503
[37] Kingman, J. F. C. (1993). Poisson Processes . Oxford Univ. Press, New York. · Zbl 0771.60001
[38] Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Z. Wahrsch. Verw. Gebiete 59 55–66. · Zbl 0482.62078
[39] Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351–357. · Zbl 0557.62036
[40] 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.
[41] 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. · Zbl 0716.62043
[42] Lo, S. H. and Phadia, E. (1992). On estimation of a survival function in reliability theory based on censored data. Preprint, Columbia Univ.
[43] McCullagh, P. (1987). Tensor Methods in Statistics . Chapman and Hall, London. · Zbl 0732.62003
[44] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability . Springer, Berlin. · Zbl 0925.60001
[45] 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. · Zbl 0496.62037
[46] 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. JSTOR: · Zbl 0455.62081
[47] Prakasa Rao, B. L. S. (1970). Estimation of distributions with monotone failure rate. Ann. Math. Statist. 41 507–519. · Zbl 0214.45903
[48] Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701–1762. · Zbl 0829.62080
[49] 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. JSTOR: · Zbl 0614.62047
[50] Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85 251–267. JSTOR: · Zbl 0951.62082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.