Dynamic modeling and statistical analysis of event times. (English) Zbl 1129.62088

Summary: This review article provides an overview of recent work in modeling and analysis of recurrent events arising in engineering, reliability, public health, biomedicine and other areas. Recurrent event modeling possesses unique facets making it different and more difficult to handle than single event settings. For instance, the impact of an increasing number of event occurrences needs to be taken into account, the effects of covariates should be considered, potential association among the interevent times within a unit cannot be ignored, and the effects of performed interventions after each event occurrence need to be factored in. A recent general class of models for recurrent events which simultaneously accommodates these aspects is described. Statistical inference methods for this class of models are presented and illustrated through applications to real data sets. Some existing open research problems are described.


62N02 Estimation in survival analysis and censored data
62Pxx Applications of statistics
Full Text: DOI arXiv Euclid


[1] Aalen, O. (1978). Nonparametric inference for a family of counting processes. Ann. Statist. 6 701–726. · Zbl 0389.62025
[2] Aalen, O. and Husebye, E. (1991). Statistical analysis of repeated events forming renewal processes. Statistics in Medicine 10 1227–1240.
[3] Agustin, M. A. and Peña, E. A. (1999). A dynamic competing risks model. Probab. Engrg. Inform. Sci. 13 333–358. · Zbl 0982.62083
[4] Agustin, Z. and Peña, E. (2005). A basis approach to goodness-of-fit testing in recurrent event models. J. Statist. Plann. Inference 133 285–303. · Zbl 1063.62139
[5] Andersen, P., Borgan, Ø., Gill, R. and Keiding, N. (1993). Statistical Models Based on Counting Processes . Springer, New York. · Zbl 0769.62061
[6] Andersen, P. and Gill, R. (1982). Cox’s regression model for counting processes: A large sample study. Ann. Statist. 10 1100–1120. · Zbl 0526.62026
[7] Barlow, R. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing : Probability Models . Holt, Rinehart and Winston, New York. · Zbl 0379.62080
[8] Baxter, L., Kijima, M. and Tortorella, M. (1996). A point process model for the reliability of a maintained system subject to general repair. Comm. Statist. Stochastic Models 12 37–65. · Zbl 0866.60075
[9] Block, H., Borges, W. and Savits, T. (1985). Age-dependent minimal repair. J. Appl. Probab. 22 370–385. JSTOR: · Zbl 0564.60084
[10] Brown, M. and Proschan, F. (1983). Imperfect repair. J. Appl. Probab. 20 851–859. JSTOR: · Zbl 0526.60080
[11] Cox, D. (1972). Regression models and life-tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187–220. JSTOR: · Zbl 0243.62041
[12] Dempster, A., Laird, N. and Rubin, D. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1–38. JSTOR: · Zbl 0364.62022
[13] Dorado, C., Hollander, M. and Sethuraman, J. (1997). Nonparametric estimation for a general repair model. Ann. Statist. 25 1140–1160. · Zbl 0937.62103
[14] Fleming, T. and Harrington, D. (1991). Counting Processes and Survival Analysis . Wiley, New York. · Zbl 0727.62096
[15] Gail, M., Santner, T. and Brown, C. (1980). An analysis of comparative carcinogenesis experiments based on multiple times to tumor. Biometrics 36 255–266. JSTOR: · Zbl 0463.62098
[16] Gill, R. (1980). Censoring and Stochastic Integrals . Mathematisch Centrum, Amsterdam. · Zbl 0456.62003
[17] Gill, R. D. (1981). Testing with replacement and the product-limit estimator. Ann. Statist. 9 853–860. · Zbl 0468.62039
[18] Gill, R. D. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 1501–1555. · Zbl 0718.60087
[19] González, J., Peña, E. and Slate, E. (2005). Modelling intervention effects after cancer relapses. Statistics in Medicine 24 3959–3975.
[20] Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18 1259–1294. · Zbl 0711.62033
[21] Hollander, M. and Peña, E. A. (1995). Dynamic reliability models with conditional proportional hazards. Lifetime Data Anal. 1 377–401. · Zbl 0859.62089
[22] Jelinski, Z. and Moranda, P. (1972). Software reliability research. In Statistical Computer Performance Evaluation (W. Freiberger, ed.) 465–484. Academic Press, New York.
[23] Kijima, M. (1989). Some results for repairable systems with general repair. J. Appl. Probab. 26 89–102. JSTOR: · Zbl 0671.60080
[24] Kumar, U. and Klefsjö, B. (1992). Reliability analysis of hydraulic systems of LHD machines using the power law process model. Reliability Engineering and System Safety 35 217–224.
[25] Kvam, P. and Peña, E. (2005). Estimating load-sharing properties in a dynamic reliability system. J. Amer. Statist. Assoc. 100 262–272. · Zbl 1117.62375
[26] Last, G. and Szekli, R. (1998). Asymptotic and monotonicity properties of some repairable systems. Adv. in Appl. Probab. 30 1089–1110. · Zbl 0933.60098
[27] Lawless, J. (1987). Regression methods for Poisson process data. J. Amer. Statist. Assoc. 82 808–815. JSTOR: · Zbl 0657.62103
[28] Lenglart, E. (1977). Relation de domination entre deux processus. Ann. Inst. H. Poincaré Sect. B ( N.S. ) 13 171–179. · Zbl 0373.60054
[29] Lindqvist, B. (1999). Repairable systems with general repair. In Proc. Tenth European Conference on Safety and Reliability, ESREL ’ 99 (G. Schueller and P. Kafka, eds.) 43–48. Balkema, Rotterdam.
[30] Lindqvist, B. H., Elvebakk, G. and Heggland, K. (2003). The trend-renewal process for statistical analysis of repairable systems. Technometrics 45 31–44.
[31] Neyman, J. (1937). “Smooth” test for goodness of fit. Skand. Aktuarietidskrift 20 149–199. · Zbl 0018.03403
[32] Nielsen, G., Gill, R., Andersen, P. and Sørensen, T. (1992). A counting process approach to maximum likelihood estimation in frailty models. Scand. J. Statist. 19 25–43. · Zbl 0747.62093
[33] Peña, E. A. (1998a). Smooth goodness-of-fit tests for composite hypothesis in hazard based models. Ann. Statist. 26 1935–1971. · Zbl 0934.62023
[34] Peña, E. A. (1998b). Smooth goodness-of-fit tests for the baseline hazard in Cox’s proportional hazards model. J. Amer. Statist. Assoc. 93 673–692. JSTOR: · Zbl 0953.62043
[35] Peña, E. and Hollander, M. (2004). Models for recurrent events in reliability and survival analysis. In Mathematical Reliability : An Expository Perspective (R. Soyer, T. Mazzuchi and N. Singpurwalla, eds.) 105–123. Kluwer, Boston.
[36] Peña, E. and Slate, E. (2005). Dynamic modeling in reliability and survival analysis. In Modern Statistical and Mathematical Methods in Reliability (A. Wilson, N. Limnios, S. Keller-McNulty and Y. Armijo, eds.) 55–71. World Scientific, Singapore. · Zbl 1082.62098
[37] Peña, E., Slate, E. and González, J. (2007). Semiparametric inference for a general class of models for recurrent events. J. Statist. Plann. Inference . · Zbl 1332.62118
[38] Peña, E. A., Strawderman, R. L. and Hollander, M. (2000). A weak convergence result relevant in recurrent and renewal models. In Recent Advances in Reliability Theory ( Bordeaux , 2000 ) (N. Limnios and M. Nikulin, eds.) 493–514. Birkhäuser, Boston. · Zbl 0981.60017
[39] Peña, E. A., Strawderman, R. L. and Hollander, M. (2001). Nonparametric estimation with recurrent event data. J. Amer. Statist. Assoc. 96 1299–1315. JSTOR: · Zbl 1073.62566
[40] Prentice, R., Williams, B. and Peterson, A. (1981). On the regression analysis of multivariate failure time data. Biometrika 68 373–379. JSTOR: · Zbl 0465.62100
[41] Rebolledo, R. (1980). Central limit theorems for local martingales. Z. Wahrsch. Verw. Gebiete 51 269–286. · Zbl 0432.60027
[42] Sellke, T. (1988). Weak convergence of the Aalen estimator for a censored renewal process. In Statistical Decision Theory and Related Topics IV (S. Gupta and J. Berger, eds.) 2 183–194. Springer, New York. · Zbl 0649.62020
[43] Singpurwalla, N. D. (1995). Survival in dynamic environments. Statist. Sci. 10 86–103. · Zbl 1148.62314
[44] Stadje, W. and Zuckerman, D. (1991). Optimal maintenance strategies for repairable systems with general degree of repair. J. Appl. Probab. 28 384–396. JSTOR: · Zbl 0731.60080
[45] Stocker, R. (2004). A general class of parametric models for recurrent event data. Ph.D. dissertation, Univ. South Carolina.
[46] Stocker, R. and Peña, E. (2007). A general class of parametric models for recurrent event data. Technometrics .
[47] Therneau, T. and Grambsch, P. (2000). Modeling Survival Data : Extending the Cox Model . Springer, New York. · Zbl 0958.62094
[48] Wang, M.-C. and Chang, S.-H. (1999). Nonparametric estimation of a recurrent survival function. J. Amer. Statist. Assoc. 94 146–153. JSTOR: · Zbl 0999.62079
[49] Wei, L., Lin, D. and Weissfeld, L. (1989). Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J. Amer. Statist. Assoc. 84 1065–1073. JSTOR:
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