## Bernoulli

• Bernoulli
• Volume 23, Number 4B (2017), 3437-3468.

### Efficiency and bootstrap in the promotion time cure model

#### Abstract

In this paper, we consider a semiparametric promotion time cure model and study the asymptotic properties of its nonparametric maximum likelihood estimator (NPMLE). First, by relying on a profile likelihood approach, we show that the NPMLE may be computed by a single maximization over a set whose dimension equals the dimension of the covariates plus one. Next, using $Z$-estimation theory for semiparametric models, we derive the asymptotics of both the parametric and nonparametric components of the model and show their efficiency. We also express the asymptotic variance of the estimator of the parametric component. Since the variance is difficult to estimate, we develop a weighted bootstrap procedure that allows for a consistent approximation of the asymptotic law of the estimators. As in the Cox model, it turns out that suitable tools are the martingale theory for counting processes and the infinite dimensional $Z$-estimation theory. Finally, by means of simulations, we show the accuracy of the bootstrap approximation.

#### Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3437-3468.

Dates
Revised: January 2016
First available in Project Euclid: 23 May 2017

https://projecteuclid.org/euclid.bj/1495505098

Digital Object Identifier
doi:10.3150/16-BEJ852

Mathematical Reviews number (MathSciNet)
MR3654812

Zentralblatt MATH identifier
1384.62117

#### Citation

Portier, François; El Ghouch, Anouar; Van Keilegom, Ingrid. Efficiency and bootstrap in the promotion time cure model. Bernoulli 23 (2017), no. 4B, 3437--3468. doi:10.3150/16-BEJ852. https://projecteuclid.org/euclid.bj/1495505098

#### References

• [1] 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.
• [2] Ash, R.B. (1972). Real Analysis and Probability. New York: Academic Press. Probability and Mathematical Statistics, No. 11.
• [3] Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Series in the Mathematical Sciences. Baltimore, MD: Johns Hopkins Univ. Press.
• [4] Breslow, N. (1972). Contribution to the discussion of the paper by D.R. Cox entitled: “Regression models and life-tables.” J. Roy. Statist. Soc. Ser. B 34 187–220.
• [5] Chen, M.-H., Ibrahim, J.G. and Sinha, D. (1999). A new Bayesian model for survival data with a surviving fraction. J. Amer. Statist. Assoc. 94 909–919.
• [6] Cheng, G. and Huang, J.Z. (2010). Bootstrap consistency for general semiparametric $M$-estimation. Ann. Statist. 38 2884–2915.
• [7] Cox, D.R. (1972). Regression models and life-tables. J. Roy. Statist. Soc. Ser. B 34 187–220.
• [8] Davison, A.C. and Hinkley, D.V. (1997). Bootstrap Methods and Their Application. Cambridge Series in Statistical and Probabilistic Mathematics 1. Cambridge: Cambridge Univ. Press.
• [9] Dudley, R.M. (1992). Fréchet differentiability, $p$-variation and uniform Donsker classes. Ann. Probab. 20 1968–1982.
• [10] Dupuy, J.-F., Grama, I. and Mesbah, M. (2006). Asymptotic theory for the Cox model with missing time-dependent covariate. Ann. Statist. 34 903–924.
• [11] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
• [12] Fleming, T.R. and Harrington, D.P. (1991). Counting Processes and Survival Analysis. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
• [13] Ibrahim, J.G., Chen, M.-H. and Sinha, D. (2001). Bayesian semiparametric models for survival data with a cure fraction. Biometrics 57 383–388.
• [14] Kosorok, M.R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics. New York: Springer.
• [15] Kosorok, M.R., Lee, B.L. and Fine, J.P. (2004). Robust inference for univariate proportional hazards frailty regression models. Ann. Statist. 32 1448–1491.
• [16] Lo, A.Y. (1993). A Bayesian bootstrap for censored data. Ann. Statist. 21 100–123.
• [17] Lu, W. (2008). Maximum likelihood estimation in the proportional hazards cure model. Ann. Inst. Statist. Math. 60 545–574.
• [18] Ma, Y. and Yin, G. (2008). Cure rate model with mismeasured covariates under transformation. J. Amer. Statist. Assoc. 103 743–756.
• [19] Murphy, S.A. (1994). Consistency in a proportional hazards model incorporating a random effect. Ann. Statist. 22 712–731.
• [20] Murphy, S.A. (1995). Asymptotic theory for the frailty model. Ann. Statist. 23 182–198.
• [21] Murphy, S.A., Rossini, A.J. and van der Vaart, A.W. (1997). Maximum likelihood estimation in the proportional odds model. J. Amer. Statist. Assoc. 92 968–976.
• [22] Murphy, S.A. and van der Vaart, A.W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95 449–485.
• [23] Præstgaard, J. and Wellner, J.A. (1993). Exchangeably weighted bootstraps of the general empirical process. Ann. Probab. 21 2053–2086.
• [24] Ritov, Y. and Wellner, J.A. (1988). Censoring, martingales, and the Cox model. In Statistical Inference from Stochastic Processes (Ithaca, NY, 1987). Contemp. Math. 80 191–219. Providence, RI: Amer. Math. Soc.
• [25] Sasieni, P. (1992). Information bounds for the conditional hazard ratio in a nested family of regression models. J. Roy. Statist. Soc. Ser. B 54 617–635.
• [26] Tsodikov, A. (1998). Asymptotic efficiency of a proportional hazards model with cure. Statist. Probab. Lett. 39 237–244.
• [27] Tsodikov, A. (1998). A proportional hazards model taking account of long-term survivors. Biometrics 54 1508–1516.
• [28] Tsodikov, A. (2001). Estimation of survival based on proportional hazards when cure is a possibility. Math. Comput. Modelling 33 1227–1236.
• [29] Tsodikov, A.D., Ibrahim, J.G. and Yakovlev, A.Y. (2003). Estimating cure rates from survival data: An alternative to two-component mixture models. J. Amer. Statist. Assoc. 98 1063–1078.
• [30] van der Vaart, A.W. (1995). Efficiency of infinite-dimensional $M$-estimators. Stat. Neerl. 49 9–30.
• [31] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press.
• [32] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer.
• [33] Wellner, J.A. and Zhan, Y. (1996). Bootstrapping $Z$-estimators. Technical Report 308. Univ. Washington, Dept. Statistics.
• [34] Yakovlev, A. and Tsodikov, A. (1996). Stochastic Models of Tumor Latency and Their Biostatistical Applications. Singapore: World Scientific.
• [35] Zeng, D., Yin, G. and Ibrahim, J.G. (2006). Semiparametric transformation models for survival data with a cure fraction. J. Amer. Statist. Assoc. 101 670–684.