The Annals of Applied Probability

Linear and quadratic functionals of random hazard rates: An asymptotic analysis

Giovanni Peccati and Igor Prünster

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Abstract

A popular Bayesian nonparametric approach to survival analysis consists in modeling hazard rates as kernel mixtures driven by a completely random measure. In this paper we derive asymptotic results for linear and quadratic functionals of such random hazard rates. In particular, we prove central limit theorems for the cumulative hazard function and for the path-second moment and path-variance of the hazard rate. Our techniques are based on recently established criteria for the weak convergence of single and double stochastic integrals with respect to Poisson random measures. The findings are illustrated by considering specific models involving kernels and random measures commonly exploited in practice. Our abstract results are of independent theoretical interest and can be applied to other areas dealing with Lévy moving average processes. The strictly Bayesian analysis is further explored in a companion paper, where our results are extended to accommodate posterior analysis.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 5 (2008), 1910-1943.

Dates
First available in Project Euclid: 30 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1225372956

Digital Object Identifier
doi:10.1214/07-AAP509

Mathematical Reviews number (MathSciNet)
MR2462554

Zentralblatt MATH identifier
1153.60028

Subjects
Primary: 60G57: Random measures 62G20: Asymptotic properties

Keywords
Asymptotics Bayesian nonparametrics central limit theorem completely random measure multiple Wiener–Itô integral path-variance random hazard rate survival analysis

Citation

Peccati, Giovanni; Prünster, Igor. Linear and quadratic functionals of random hazard rates: An asymptotic analysis. Ann. Appl. Probab. 18 (2008), no. 5, 1910--1943. doi:10.1214/07-AAP509. https://projecteuclid.org/euclid.aoap/1225372956


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References

  • [1] Aalen, O. (1978). Nonparametric inference for a family of counting processes. Ann. Statist. 6 701–726.
    Mathematical Reviews (MathSciNet): MR491547
    Zentralblatt MATH: 0389.62025
    Digital Object Identifier: doi:10.1214/aos/1176344247
    Project Euclid: euclid.aos/1176344247
  • [2] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241.
    Mathematical Reviews (MathSciNet): MR1841412
    Zentralblatt MATH: 0983.60028
    Digital Object Identifier: doi:10.1111/1467-9868.00282
    JSTOR: links.jstor.org
  • [3] Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. in Appl. Probab. 31 929–953.
    Mathematical Reviews (MathSciNet): MR1747450
    Zentralblatt MATH: 0957.60055
    Digital Object Identifier: doi:10.1239/aap/1029955251
    Project Euclid: euclid.aap/1029955251
  • [4] Cifarelli, D. M. and Melilli, E. (2000). Some new results for Dirichlet priors. Ann. Statist. 28 1390–1413.
    Mathematical Reviews (MathSciNet): MR1805789
    Zentralblatt MATH: 1105.62303
    Digital Object Identifier: doi:10.1214/aos/1015957399
    Project Euclid: euclid.aos/1015957399
  • [5] Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18 429–442.
    Mathematical Reviews (MathSciNet): MR1041402
    Digital Object Identifier: doi:10.1214/aos/1176347509
    Project Euclid: euclid.aos/1176347509
  • [6] De Blasi, P. and Hjort, N. L. (2007). Bayesian survival analysis in proportional hazard models with logistic relative risk. Scand. J. Statist. 34 229–257.
    Mathematical Reviews (MathSciNet): MR2325252
    Digital Object Identifier: doi:10.1111/j.1467-9469.2006.00543.x
  • [7] De Blasi, P., Peccati, G. and Prünster, I. (2007). Asymptotics for posterior hazards. Technical report. Submitted.
  • [8] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
    Mathematical Reviews (MathSciNet): MR950166
  • [9] Deheuvels, P., Peccati, G. and Yor, M. (2006). On quadratic functionals of the Brownian sheet and related processes. Stochastic Process. Appl. 116 493–538.
    Mathematical Reviews (MathSciNet): MR2199561
    Zentralblatt MATH: 1090.60020
    Digital Object Identifier: doi:10.1016/j.spa.2005.10.004
  • [10] 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
  • [11] Drăghici, L. and Ramamoorthi, R. V. (2003). Consistency of Dykstra–Laud priors. Sankhyā 65 464–481.
  • [12] 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
  • [13] Epifani, I., Lijoi, A. and Prünster, I. (2003). Exponential functionals and means of neutral-to-the-right priors. Biometrika 90 791–808.
  • [14] Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615–629.
    Mathematical Reviews (MathSciNet): MR438568
    Zentralblatt MATH: 0286.62008
    Digital Object Identifier: doi:10.1214/aos/1176342752
    Project Euclid: euclid.aos/1176342752
  • [15] Ferguson, T. S. and Phadia, E. G. (1979). Bayesian nonparametric estimation based on censored data. Ann. Statist. 7 163–186.
    Mathematical Reviews (MathSciNet): MR515691
    Zentralblatt MATH: 0401.62031
    Digital Object Identifier: doi:10.1214/aos/1176344562
    Project Euclid: euclid.aos/1176344562
  • [16] Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1992245
  • [17] 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
  • [18] 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
  • [19] Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96 161–173.
    Mathematical Reviews (MathSciNet): MR1952729
    Zentralblatt MATH: 1014.62006
    Digital Object Identifier: doi:10.1198/016214501750332758
    JSTOR: links.jstor.org
  • [20] 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
  • [21] James, L. F. (2003). Bayesian calculus for gamma processes with applications to semiparametric intensity models. Sankhyā 65 179–206.
    Mathematical Reviews (MathSciNet): MR2016784
  • [22] 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
  • [23] James, L. F. (2006). Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34 416–440.
    Mathematical Reviews (MathSciNet): MR2275248
    Zentralblatt MATH: 1091.62012
    Digital Object Identifier: doi:10.1214/009053605000000732
    Project Euclid: euclid.aos/1146576269
  • [24] Kalbfleisch, J. D. (1978). Non-parametric Bayesian analysis of survival time data. J. Roy. Statist. Soc. Ser. B 40 214–221.
    Mathematical Reviews (MathSciNet): MR517442
    JSTOR: links.jstor.org
  • [25] 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
  • [26] Kim, Y. and Lee, J. (2003). Bayesian analysis of proportional hazard models. Ann. Statist. 31 493–511.
    Mathematical Reviews (MathSciNet): MR1983539
    Zentralblatt MATH: 1053.62036
    Digital Object Identifier: doi:10.1214/aos/1051027878
    Project Euclid: euclid.aos/1051027878
  • [27] 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
  • [28] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. Oxford Univ. Press, New York.
    Mathematical Reviews (MathSciNet): MR1207584
  • [29] Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
  • [30] Lijoi, A., Mena, R. H. and Prünster, I. (2005). Hierarchical mixture modeling with normalized inverse-Gaussian priors. J. Amer. Statist. Assoc. 100 1278–1291.
  • [31] Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Z. Wahrsch. Verw. Gebiete 59 55–66.
    Mathematical Reviews (MathSciNet): MR643788
  • [32] 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
  • [33] 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
  • [34] 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
  • [35] Nualart, D. and Vives, J. (1990). Anticipative calculus for the Poisson process based on the Fock space. In Séminaire de Probabilités XXIV, 1988/89. Lecture Notes in Mathematics 1426 154–165. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1071538
    Zentralblatt MATH: 0701.60048
    Digital Object Identifier: doi:10.1007/BFb0083763
  • [36] Peccati, G. (2001). On the convergence of multiple random integrals. Studia Sci. Math. Hungar. 37 429–470.
    Mathematical Reviews (MathSciNet): MR1874695
  • [37] Peccati, G. and Taqqu, M. S. (2007). Limit theorems for multiple stochastic integrals. Preprint. Available at http://geocities.com/giovannipeccati/.
    Mathematical Reviews (MathSciNet): MR2461790
    Zentralblatt MATH: 1162.60311
  • [38] Peccati, G. and Taqqu, M. S. (2007). Central limit theorems for double Poisson integrals. Bernoulli. To appear.
    Mathematical Reviews (MathSciNet): MR2537812
    Digital Object Identifier: doi:10.3150/08-BEJ123
    Project Euclid: euclid.bj/1219669630
  • [39] Peccati, G. and Taqqu, M. S. (2007). Stable convergence of generalized L2 stochastic integrals and the principle of conditioning. Electron. J. Probab. 12 no. 15, 447–480 (electronic).
    Mathematical Reviews (MathSciNet): MR2299924
    Zentralblatt MATH: 1139.60024
  • [40] Peccati, G. and Tudor, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics 1857 247–262. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR2126978
    Zentralblatt MATH: 1063.60027
  • [41] Regazzini, E., Guglielmi, A. and Di Nunno, G. (2002). Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Ann. Statist. 30 1376–1411.
    Mathematical Reviews (MathSciNet): MR1936323
    Zentralblatt MATH: 1018.62011
    Digital Object Identifier: doi:10.1214/aos/1035844980
    Project Euclid: euclid.aos/1035844980
  • [42] Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist. 31 560–585.
  • [43] Rota, G.-C. and Wallstrom, T. 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
  • [44] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. (Translated from the Japanese.)
    Mathematical Reviews (MathSciNet): MR1739520
  • [45] Surgailis, D. (1984). On multiple Poisson stochastic integrals and associated Markov semigroups. Probab. Math. Statist. 3 217–239.
    Mathematical Reviews (MathSciNet): MR764148
    Zentralblatt MATH: 0548.60058
  • [46] Surgailis, D. (2003). Non-CLTs: U-statistics, multinomial formula and approximations of multiple Itô–Wiener integrals. In Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M.S. Taqqu, eds.) 129–142. Birkhäuser, Boston, MA.
  • [47] Walker, S. and Damien, P. (1998). A full Bayesian non-parametric analysis involving a neutral to the right process. Scand. J. Statist. 25 669–680.
    Mathematical Reviews (MathSciNet): MR1666808
    Digital Object Identifier: doi:10.1111/1467-9469.00128
  • [48] Walker, S. and Muliere, P. (1997). Beta-Stacy processes and a generalization of the Pólya-urn scheme. Ann. Statist. 25 1762–1780.
  • [49] Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85 251–267.
    Mathematical Reviews (MathSciNet): MR1649114
    Zentralblatt MATH: 0951.62082
    Digital Object Identifier: doi:10.1093/biomet/85.2.251
    JSTOR: links.jstor.org
  • [50] Wolpert, R. L. and Taqqu, R. L. (2005). Fractional Ornstein–Uhlenbeck Lévy processes and the Telecom process. Signal Process. 85 1523–1545.

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