• Bernoulli
  • Volume 23, Number 4B (2017), 3650-3684.

The failure of the profile likelihood method for a large class of semi-parametric models

Eric Beutner, Laurent Bordes, and Laurent Doyen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector $\mathbf{{\theta}}$, say, exceeds the profile likelihood function evaluated at the parameter vector $\tilde{\boldsymbol {\theta}}$, say, with probability $p$. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we will provide an example where the profile likelihood function is monotone with probability one regardless of the true data generating process. We also discuss the relation of our results to other semi-parametric models like the accelerated failure time model and Cox’s proportional hazards model.

Article information

Bernoulli, Volume 23, Number 4B (2017), 3650-3684.

Received: February 2015
Revised: April 2016
First available in Project Euclid: 23 May 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

accelerated failure time model Cox’s proportional hazards model effective age process profile likelihood inference recurrent event data semi-parametric statistical model virtual age process


Beutner, Eric; Bordes, Laurent; Doyen, Laurent. The failure of the profile likelihood method for a large class of semi-parametric models. Bernoulli 23 (2017), no. 4B, 3650--3684. doi:10.3150/16-BEJ861.

Export citation


  • [1] Aalen, O.O., Borgan, Ø. and Gjessing, H.K. (2008). Survival and Event History Analysis: A Process Point of View. Statistics for Biology and Health. New York: Springer.
  • [2] Adekpedjou, A. and Stocker, R. (2015). A general class of semiparametric models for recurrent event data. J. Statist. Plann. Inference 156 48–63.
  • [3] Andersen, P.K., Borgan, Ø., Gill, R.D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer Series in Statistics. New York: Springer.
  • [4] Andersen, P.K. and Gill, R.D. (1982). Cox’s regression model for counting processes: A large sample study. Ann. Statist. 10 1100–1120.
  • [5] Barndorff-Nielsen, O.E. (1988). Parametric Statistical Models and Likelihood. Lecture Notes in Statistics 50. New York: Springer.
  • [6] Barndorff-Nielsen, O.E. and Cox, D.R. (1994). Inference and Asymptotics. Monographs on Statistics and Applied Probability 52. London: Chapman & Hall.
  • [7] Braekers, R. and Veraverbeke, N. (2005). Cox’s regression model under partially informative censoring. Comm. Statist. Theory Methods 34 1793–1811.
  • [8] Breslow, N., McNeney, B. and Wellner, J.A. (2003). Large sample theory for semiparametric regression models with two-phase, outcome dependent sampling. Ann. Statist. 31 1110–1139.
  • [9] Claeskens, G. and Carroll, R.J. (2007). An asymptotic theory for model selection inference in general semiparametric problems. Biometrika 94 249–265.
  • [10] Cook, R.J. and Lawless, J. (2007). The Statistical Analysis of Recurrent Events. New York: Springer Science + Business Media.
  • [11] Davison, A.C. (2003). Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics 11. Cambridge: Cambridge Univ. Press.
  • [12] Dorado, C., Hollander, M. and Sethuraman, J. (1997). Nonparametric estimation for a general repair model. Ann. Statist. 25 1140–1160.
  • [13] Doyen, L. and Gaudoin, O. (2004). Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliab. Eng. Syst. Saf. 84 45–56.
  • [14] Fraser, D.A.S. (2003). Likelihood for component parameters. Biometrika 90 327–339.
  • [15] Gärtner, M. (2003). Estimation in a general repair model based on left-truncated data. Scand. J. Stat. 30 61–73.
  • [16] Gill, R.D. (1981). Testing with replacement and the product limit estimator. Ann. Statist. 9 853–860.
  • [17] González, J.R., Peña, E.A. and Slate, E.H. (2005). Modelling intervention effects after cancer relapses. Stat. Med. 24 3959–3975.
  • [18] Hirose, Y. (2011). Efficiency of profile likelihood in semi-parametric models. Ann. Inst. Statist. Math. 63 1247–1275.
  • [19] Hollander, M. and Sethuraman, J. (2004). Nonparametric methods for repair models. In Advances in Survival Analysis. Handbook of Statist. 23 747–763. Amsterdam: Elsevier.
  • [20] Huang, C.-Y., Qin, J. and Follmann, D.A. (2012). A maximum pseudo-profile likelihood estimator for the Cox model under length-biased sampling. Biometrika 99 199–210.
  • [21] Jacod, J. (1975). Multivariate point processes: Predictable projection, Radon–Nikodým derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235–253.
  • [22] Johansen, S. (1983). An extension of Cox’s regression model. Int. Stat. Rev. 51 165–174.
  • [23] Kijima, M. (1989). Some results for repairable systems with general repair. J. Appl. Probab. 26 89–102.
  • [24] Kijima, M., Morimura, H. and Suzuki, Y. (1988). Periodical replacement problem without assuming minimal repair. European J. Oper. Res. 37 194–203.
  • [25] Last, G. and Szekli, R. (1998). Asymptotic and monotonicity properties of some repairable systems. Adv. in Appl. Probab. 30 1089–1110.
  • [26] Lawless, J.F. (1995). The analysis of recurrent events for multiple subjects. J. R. Stat. Soc. Ser. C. Appl. Stat. 44 487–498.
  • [27] Lindqvist, B.H. (2006). On the statistical modeling and analysis of repairable systems. Statist. Sci. 21 532–551.
  • [28] McCullagh, P. and Tibshirani, R. (1990). A simple method for the adjustment of profile likelihoods. J. R. Stat. Soc. Ser. B. Stat. Methodol. 52 325–344.
  • [29] Murphy, S.A. and van der Vaart, A.W. (2000). On profile likelihood (with discussion). J. Amer. Statist. Assoc. 95 449–485.
  • [30] Nelson, W.B. (2003). Recurrent Events Data Analysis for Product Repairs, Disease Recurrences, and Other Applications. ASA-SIAM Series on Statistics and Applied Probability 10. Philadelphia, PA: SIAM.
  • [31] Peña, E.A. (2006). Dynamic modeling and statistical analysis of event times. Statist. Sci. 21 487–500.
  • [32] Peña, E.A. (2016). Asymptotics for a class of dynamic recurrent event models. J. Nonparametr. Stat. To appear.
  • [33] Peña, E.A. and Hollander, M. (2004). Models for recurrent events in reliability and survival analysis. In Mathematical Reliability: An Expository Perspective. Internat. Ser. Oper. Res. Management Sci. 67 105–123. Boston, MA: Kluwer Academic.
  • [34] Peña, E.A., Slate, E.H. and González, J.R. (2007). Semiparametric inference for a general class of models for recurrent events. J. Statist. Plann. Inference 137 1727–1747.
  • [35] Peña, E.A., Strawderman, R.L. and Hollander, M. (2001). Nonparametric estimation with recurrent event data. J. Amer. Statist. Assoc. 96 1299–1315.
  • [36] Reid, N. (2013). Aspects of likelihood inference. Bernoulli 19 1404–1418.
  • [37] Scott, A.J. and Wild, C.J. (1997). Fitting regression models to case–control data by maximum likelihood. Biometrika 84 57–71.
  • [38] Sellke, T. (1988). Weak convergence of the Aalen estimator for a censored renewal process. In Statistical Decision Theory and Related Topics, IV, Vol. 2 (West Lafayette, Ind., 1986) (S. Gupta and J. Berger, eds.) 183–194. New York: Springer.
  • [39] Sellke, T. and Siegmund, D. (1983). Sequential analysis of the proportional hazards model. Biometrika 70 315–326.
  • [40] Severini, T.A. (2000). Likelihood Methods in Statistics. Oxford Statistical Science Series 22. Oxford: Oxford Univ. Press.
  • [41] Severini, T.A. and Wong, W.H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist. 20 1768–1802.
  • [42] Slud, E.V. and Vonta, F. (2005). Efficient semiparametric estimators via modified profile likelihood. J. Statist. Plann. Inference 129 339–367.
  • [43] Xu, R., Vaida, F. and Harrington, D.P. (2009). Using profile likelihood for semiparametric model selection with application to proportional hazards mixed models. Statist. Sinica 19 819–842.
  • [44] Zeng, D. and Lin, D.Y. (2007). Efficient estimation for the accelerated failure time model. J. Amer. Statist. Assoc. 102 1387–1396.
  • [45] Zeng, D. and Lin, D.Y. (2010). A general asymptotic theory for maximum likelihood estimation in semiparametric regression models with censored data. Statist. Sinica 20 871–910.