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Van der Laan, Mark J.; Jewell, Nicholas P.

Current status and right-censored data structures when observing a marker at the censoring time. (English) Zbl 1039.62095

Ann. Stat. 31, No. 2, 512-535 (2003).
Summary: 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.

Cited in 1 Review
Cited in 4 Documents

MSC:

62N02 Estimation in survival analysis and censored data
62N01 Censored data models
62G05 Nonparametric estimation
62F12 Asymptotic properties of parametric estimators
62G07 Density estimation

Keywords:

asymptotically linear estimator; asymptotically efficient estimator; current status data; right-censored data; isotonic regression
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References:

[1] BARLOW, R. E., BARTHOLOMEW, D. J., BREMNER, J. M. and BRUNK, H. D. (1972). Statistical Inference under Order Restrictions. Wiley, New York. · Zbl 0246.62038
[2] 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. · Zbl 0816.62032
[3] 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.
[4] 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.
[5] DINSE, G. E. and LAGAKOS, S. W. (1982). Nonparametric estimation of lifetime and disease onset distributions from incomplete observations. Biometrics 38 921-932. · Zbl 0504.62099 · doi:10.2307/2529872
[6] 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. · Zbl 0918.62003
[7] 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.) URL:
[8] GROENEBOOM, P. and WELLNER, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel. · Zbl 0757.62017
[9] 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. · Zbl 0832.62029 · doi:10.1111/j.1467-9574.1995.tb01462.x
[10] 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. · Zbl 0793.62066 · doi:10.2307/2291196
[11] JEWELL, N. P. and SHIBOSKI, S. C. (1990). Statistical analysis of HIV infectivity based on partner studies. Biometrics 46 1133-1150.
[12] JEWELL, N. P. and VAN DER LAAN, M. J. (1995). Generalizations of current status data with applications. Lifetime Data Analy sis 1 101-109. · Zbl 0822.62093 · doi:10.1007/BF00985261
[13] JONGBLOED, G. (1995). Three statistical inverse problems. Ph.D. dissertation, Delft Univ. Technology.
[14] KEIDING, N. (1991). Age-specific incidence and prevalence: A statistical perspective (with discussion). J. Roy. Statist. Soc. Ser. A 154 371-412. JSTOR: · Zbl 1002.62504 · doi:10.2307/2983150
[15] 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.
[16] SUN, J. and KALBFLEISCH, J. D. (1993). The analysis of current status data on point processes. J. Amer. Statist. Assoc. 88 1449-1454. JSTOR: · Zbl 0793.62067 · doi:10.2307/2291291
[17] 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.
[18] 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. · Zbl 0882.62109 · doi:10.1093/biomet/84.3.539
[19] BERKELEY, CALIFORNIA 94720 E-MAIL: laan@stat.berkeley.edu
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