## The Annals of Statistics

### Current status and right-censored data structures when observing a marker at the censoring time

#### Abstract

We study nonparametric estimation with two types of data structures. In the first data structure n i.i.d. copies of $(C,N(C))$ are observed, where N is a finite state counting process jumping at time-variables of interest and C a random monitoring time. In the second data structure n i.i.d. copies of $(C\wedge T,I(T\leq C),N (C\wedge T))$ are observed, where N is a counting process with a final jump at time T (e.g., death). This data structure includes observing right-censored data on T and a marker variable at the censoring time.

In these data structures, easy to compute estimators, namely (weighted)-pool-adjacent-violator estimators for the marginal distributions of the unobservable time variables, and the Kaplan-Meier estimator for the time T till the final observable event, are available. These estimators ignore seemingly important information in the data. In this paper we prove that, at many continuous data generating distributions the ad hoc estimators yield asymptotically efficient estimators of $\sqrt{n}$-estimable parameters.

#### Article information

Source
Ann. Statist., Volume 31, Number 2 (2003), 512-535.

Dates
First available in Project Euclid: 22 April 2003

https://projecteuclid.org/euclid.aos/1051027879

Digital Object Identifier
doi:10.1214/aos/1051027879

Mathematical Reviews number (MathSciNet)
MR1983540

Zentralblatt MATH identifier
1039.62095

Subjects
Primary: 62G07: Density estimation
Secondary: 62F12: Asymptotic properties of estimators

#### Citation

Van der Laan, Mark J.; Jewell, Nicholas P. Current status and right-censored data structures when observing a marker at the censoring time. Ann. Statist. 31 (2003), no. 2, 512--535. doi:10.1214/aos/1051027879. https://projecteuclid.org/euclid.aos/1051027879

#### References

• BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M. and BRUNK, H. D. (1972). Statistical Inference under Order Restrictions. Wiley, New York.
• BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. (1993). Efficient and Adaptive Estimation in Semi-Parametric Models. Johns Hopkins Univ. Press.
• DIAMOND, I. D. and MCDONALD, J. W. (1992). The analysis of current status data. In Demographic Applications of Event History Analy sis (J. Trussell, R. Hankinson and J. Tilton, eds.) 231- 252. Oxford Univ. Press.
• DIAMOND, I. D., MCDONALD, J. W. and SHAH, I. H. (1986). Proportional hazards models for current status data: Application to the study of differentials in age at weaning in Pakistan. Demography 23 607-620.
• DINSE, G. E. and LAGAKOS, S. W. (1982). Nonparametric estimation of lifetime and disease onset distributions from incomplete observations. Biometrics 38 921-932.
• GILL, R. D., VAN DER LAAN, M. J. and ROBINS, J. M. (1997). Coarsening at random: Characterizations, conjectures and counterexamples. Proc. First Seattle Sy mposium in Biostatistics. Lecture Notes in Statist. 123 255-294. Springer, New York.
• GROENEBOOM, P. J. (1998). Special topics course 593C: Nonparametric estimation for inverse problems: algorithms and asy mptotics. Technical Report 344, Dept. Statistics, Univ. Washington. (For related software see www.stat.washington.edu/jaw/RESEARCH/SOFTWARE/ software.list.html.)
• GROENEBOOM, P. and WELLNER, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel.
• HUANG, J. and WELLNER, J. A. (1995). Asy mptotic normality of the NPMLE of linear functionals for interval censored data, case I. Statist. Neerlandica 49 153-163.
• JEWELL, N. P., MALANI, H. M. and VITTINGHOFF, E. (1994). Nonparametric estimation for a form of doubly censored data with application to two problems in AIDS. J. Amer. Statist. Assoc. 89 7-18.
• JEWELL, N. P. and SHIBOSKI, S. C. (1990). Statistical analysis of HIV infectivity based on partner studies. Biometrics 46 1133-1150.
• JEWELL, N. P. and VAN DER LAAN, M. J. (1995). Generalizations of current status data with applications. Lifetime Data Analy sis 1 101-109.
• JONGBLOED, G. (1995). Three statistical inverse problems. Ph.D. dissertation, Delft Univ. Technology.
• KEIDING, N. (1991). Age-specific incidence and prevalence: A statistical perspective (with discussion). J. Roy. Statist. Soc. Ser. A 154 371-412.
• KODELL, R. L., SHAW, G. W. and JOHNSON, A. M. (1982). Nonparametric joint estimators for disease resistance and survival functions in survival/sacrifice experiments. Biometrics 38 43-58.
• SUN, J. and KALBFLEISCH, J. D. (1993). The analysis of current status data on point processes. J. Amer. Statist. Assoc. 88 1449-1454.
• TURNBULL, B. W. and MITCHELL, T. J. (1984). Nonparametric estimation of the distribution of time to onset for specific diseases in survival/sacrifice experiments. Biometrics 40 41-50.
• VAN DER LAAN, M. J., JEWELL, N. P. and PETERSON, D. R. (1997). Efficient estimation of the lifetime and disease onset distribution. Biometrika 84 539-554.
• BERKELEY, CALIFORNIA 94720 E-MAIL: laan@stat.berkeley.edu