Asymptotics for posterior hazards

Asymptotics for posterior hazards

Pierpaolo De Blasi, Giovanni Peccati, and Igor Prünster

Source: Ann. Statist. Volume 37, Number 4 (2009), 1906-1945.

Abstract

An important issue in survival analysis is the investigation and the modeling of hazard rates. Within a Bayesian nonparametric framework, a natural and popular approach is to model hazard rates as kernel mixtures with respect to a completely random measure. In this paper we provide a comprehensive analysis of the asymptotic behavior of such models. We investigate consistency of the posterior distribution and derive fixed sample size central limit theorems for both linear and quadratic functionals of the posterior hazard rate. The general results are then specialized to various specific kernels and mixing measures yielding consistency under minimal conditions and neat central limit theorems for the distribution of functionals.

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Primary Subjects: 62G20, 60G57

Keywords: Asymptotics; Bayesian consistency; Bayesian nonparametrics; central limit theorem; completely random measure; path-variance; random hazard rate; survival analysis

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Permanent link to this document: http://projecteuclid.org/euclid.aos/1245332836
Digital Object Identifier: doi:10.1214/08-AOS631
Zentralblatt MATH identifier: 05582014
Mathematical Reviews number (MathSciNet): MR2533475

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References

[1] Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of distributions in nonparametric problems. Ann. Statist. 27 536–561.

Mathematical Reviews (MathSciNet): MR1714718

Zentralblatt MATH: 0980.62039

Digital Object Identifier: doi:10.1214/aos/1017939142

Project Euclid: euclid.aos/1018031206

[2] Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. in Appl. Prob. 31 929–953.

Mathematical Reviews (MathSciNet): MR1747450

Zentralblatt MATH: 0957.60055

Digital Object Identifier: doi:10.1239/aap/1029955251

Project Euclid: euclid.aap/1029955251

[3] Daley, D. and Vere-Jones, D. J. (1988). An Introduction to the Theory of Point Processes. Springer, New York.

Mathematical Reviews (MathSciNet): MR950166

[4] Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183–201.

Mathematical Reviews (MathSciNet): MR373081

Digital Object Identifier: doi:10.1214/aop/1176996703

[5] Drăghici, L. and Ramamoorthi, R. V. (2003). Consistency of Dykstra-Laud priors. Sankhyā 65 464–481.

Mathematical Reviews (MathSciNet): MR2028910

[6] Dykstra, R. L. and Laud, P. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9 356–367.

Mathematical Reviews (MathSciNet): MR606619

Zentralblatt MATH: 0469.62077

Digital Object Identifier: doi:10.1214/aos/1176345401

Project Euclid: euclid.aos/1176345401

[7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 3rd ed. Wiley, New York.

Mathematical Reviews (MathSciNet): MR270403

[8] Ghosal, S., Ghosh, J. K. and Ramamoorthi, R. V. (1999). Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27 143–158.

Mathematical Reviews (MathSciNet): MR1701105

Zentralblatt MATH: 0932.62043

Digital Object Identifier: doi:10.1214/aos/1018031105

Project Euclid: euclid.aos/1018031105

[9] Ghosh, J. K. and Ramamoorthi, R. V. (1995). Consistency of Bayesian inference for survival analysis with or without censoring. In Analysis of Censored Data (Pune, 1994/1995). IMS Lecture Notes Monogr. Ser. 27 95–103. IMS, Hayward, CA.

Mathematical Reviews (MathSciNet): MR1483342

Zentralblatt MATH: 0876.62021

Digital Object Identifier: doi:10.1214/lnms/1215452215

[10] Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer, New York.

Mathematical Reviews (MathSciNet): MR1992245

[11] Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18 1259–1294.

Mathematical Reviews (MathSciNet): MR1062708

Zentralblatt MATH: 0711.62033

Digital Object Identifier: doi:10.1214/aos/1176347749

Project Euclid: euclid.aos/1176347749

[12] Ho, M.-W. (2006). A Bayes method for a monotone hazard rate via S-paths. Ann. Statist. 34 820–836.

Mathematical Reviews (MathSciNet): MR2283394

Digital Object Identifier: doi:10.1214/009053606000000047

Project Euclid: euclid.aos/1151418242

[13] 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.

Mathematical Reviews (MathSciNet): MR2054297

Zentralblatt MATH: 1089.62520

Digital Object Identifier: doi:10.1198/016214504000000179

[14] James, L. F. (2003). Bayesian calculus for gamma processes with applications to semiparametric intensity models. Sankhyā 65 179–206.

Mathematical Reviews (MathSciNet): MR2016784

[15] James, L. F. (2005). Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33 1771–1799.

Mathematical Reviews (MathSciNet): MR2166562

Digital Object Identifier: doi:10.1214/009053605000000336

Project Euclid: euclid.aos/1123250229

[16] Kabanov, Y. (1975). On extendend stochastic integrals. Teor. Verojatnost. i Primenen. 20 725–737.

Mathematical Reviews (MathSciNet): MR397877

[17] Kallenberg, O. (1986). Random Measures, 4th ed. Akademie-Verlag, Berlin.

Mathematical Reviews (MathSciNet): MR854102

[18] Kim, Y. (1999). Nonparametric Bayesian estimators for counting processes. Ann. Statist. 27 562–588.

Mathematical Reviews (MathSciNet): MR1714717

Zentralblatt MATH: 0980.62078

Digital Object Identifier: doi:10.1214/aos/1018031207

Project Euclid: euclid.aos/1018031207

[19] Kim, Y. (2003). On the posterior consistency of mixtures of Dirichlet process priors with censored data. Scand. J. Statist. 30 535–547.

Mathematical Reviews (MathSciNet): MR2002227

Digital Object Identifier: doi:10.1111/1467-9469.00347

[20] Kim, Y. and Lee, J. (2001). On posterior consistency of survival models. Ann. Statist. 29 666–686.

Mathematical Reviews (MathSciNet): MR1865336

Zentralblatt MATH: 1012.62105

Digital Object Identifier: doi:10.1214/aos/1009210685

Project Euclid: euclid.aos/1009210685

[21] Kingman, J. F. C. (1967). Completely random measures. Pacific J. Math. 21 59–78.

Mathematical Reviews (MathSciNet): MR210185

Zentralblatt MATH: 0155.23503

Project Euclid: euclid.pjm/1102992601

[22] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. Clarendon Press, New York.

Mathematical Reviews (MathSciNet): MR1207584

[23] Lijoi, A., Prünster, I. and Walker, S. G. (2008). Posterior analysis for some classes of nonparametric models. J. Nonparametr. Stat. 20 447–457.

[24] 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.

Mathematical Reviews (MathSciNet): MR1006487

Zentralblatt MATH: 0716.62043

Digital Object Identifier: doi:10.1007/BF00049393

[25] Nieto-Barajas, L. E. and Walker, S. G. (2004). Bayesian nonparametric survival analysis via Lévy driven Markov processes. Statist. Sinica 14 1127–1146.

Mathematical Reviews (MathSciNet): MR2126344

[26] Nieto-Barajas, L. E. and Walker, S. G. (2005). A semi-parametric Bayesian analysis of survival data based on Lévy-driven processes. Lifetime Data Anal. 11 529–543.

Mathematical Reviews (MathSciNet): MR2213503

Digital Object Identifier: doi:10.1007/s10985-005-5238-7

[27] Peccati, G. and Prünster, I. (2008). Linear and quadratic functionals of random hazard rates: An asymptotic analysis. Ann. Appl. Probab. 18 1910–1943.

[28] Peccati, G. and Taqqu, M. S. (2008). Central limit theorems for double Poisson integrals. Bernoulli 14 791–821.

Mathematical Reviews (MathSciNet): MR2537812

Digital Object Identifier: doi:10.3150/08-BEJ123

Project Euclid: euclid.bj/1219669630

[29] Peterson, A.V. (1977). Expressing the Kaplan–Meier estimator as a function of empirical subsurvival functions. J. Amer. Statist. Assoc. 72 854–858.

Mathematical Reviews (MathSciNet): MR471165

Zentralblatt MATH: 0372.62077

Digital Object Identifier: doi:10.2307/2286474

JSTOR: links.jstor.org

[30] Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of random measures with independent increments. Ann. Statist. 31 560–585.

[31] Rota, G.-C. and Wallstrom, C. (1997). Stochastic integrals: A combinatorial approach. Ann. Probab. 25 1257–1283.

Mathematical Reviews (MathSciNet): MR1457619

Zentralblatt MATH: 0886.60046

Digital Object Identifier: doi:10.1214/aop/1024404513

Project Euclid: euclid.aop/1024404513

[32] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.

Mathematical Reviews (MathSciNet): MR1739520

[33] Schwarz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10–26.

Mathematical Reviews (MathSciNet): MR184378

Digital Object Identifier: doi:10.1007/BF00535479

[34] Surgailis, D. (1984). On multiple Poisson integrals and associated Markov semigroups. Probab. Math. Statist. 3 217–239.

Mathematical Reviews (MathSciNet): MR764148

Zentralblatt MATH: 0548.60058

[35] Walker, S. G. (2003). On sufficient conditions for Bayesian consistency. Biometrika 90 482–488.

Mathematical Reviews (MathSciNet): MR1986664

Zentralblatt MATH: 1034.62024

Digital Object Identifier: doi:10.1093/biomet/90.2.482

[36] Walker, S. G. (2004). New approaches to Bayesian consistency. Ann. Statist. 32 2028–2043.

Mathematical Reviews (MathSciNet): MR2102501

Zentralblatt MATH: 1056.62040

Digital Object Identifier: doi:10.1214/009053604000000409

Project Euclid: euclid.aos/1098883780

[37] Wu, Y. and Ghosal, S. (2008). Kullback Leibler property of kernel mixture priors in Bayesian density estimation. Electron. J. Stat. 2 298–331.

Mathematical Reviews (MathSciNet): MR2399197

Zentralblatt MATH: 1135.62022

Digital Object Identifier: doi:10.1214/07-EJS130

Project Euclid: euclid.ejs/1210254767

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