The Annals of Statistics

Asymptotic behavior of the unconditional NPMLE of the length-biased survivor function from right censored prevalent cohort data

Masoud Asgharian and David B. Wolfson

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Right censored survival data collected on a cohort of prevalent cases with constant incidence are length-biased, and may be used to estimate the length-biased (i.e., prevalent-case) survival function. When the incidence rate is constant, so-called stationarity of the incidence, it is more efficient to use this structure for unconditional statistical inference than to carry out an analysis by conditioning on the observed truncation times. It is well known that, due to the informative censoring for prevalent cohort data, the Kaplan–Meier estimator is not the unconditional NPMLE of the length-biased survival function and the asymptotic properties of the NPMLE do not follow from any known result. We present here a detailed derivation of the asymptotic properties of the NPMLE of the length-biased survival function from right censored prevalent cohort survival data with follow-up. In particular, we show that the NPMLE is uniformly strongly consistent, converges weakly to a Gaussian process, and is asymptotically efficient. One important spin-off from these results is that they yield the asymptotic properties of the NPMLE of the incident-case survival function [see Asgharian, M’Lan and Wolfson J. Amer. Statist. Assoc. 97 (2002) 201–209], which is often of prime interest in a prevalent cohort study. Our results generalize those given by Vardi and Zhang [Ann. Statist. 20 (1992) 1022–1039] under multiplicative censoring, which we show arises as a degenerate case in a prevalent cohort setting.

Article information

Ann. Statist., Volume 33, Number 5 (2005), 2109-2131.

First available in Project Euclid: 25 November 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62N02: Estimation

Prevalent cohort censored data length-biased sampling informative censoring survival function NPMLE consistency weak convergence and asymptotic efficiency


Asgharian, Masoud; Wolfson, David B. Asymptotic behavior of the unconditional NPMLE of the length-biased survivor function from right censored prevalent cohort data. Ann. Statist. 33 (2005), no. 5, 2109--2131. doi:10.1214/009053605000000372.

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  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Asgharian, M., Wolfson, D. B. and Zhang, X. (2004). A simple criterion for the stationarity of the incidence rate from prevalent cohort studies. Technical Report 2004-01, Dept. Mathematics and Statistics, McGill Univ.
  • Asgharian, M., Wolfson, D. B. and Zhang, X. (2005). Checking stationarity of the incidence rate using prevalent cohort survival data. Statistics in Medicine. To appear.
  • Asgharian, M., M'Lan, C. E. and Wolfson, D. B. (2002). Length-biased sampling with right censoring: An unconditional approach. J. Amer. Statist. Assoc. 97 201--209.
  • Beran, R. (1977). Estimating a distribution function. Ann. Statist. 5 400--404.
  • Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore.
  • Brillinger, D. R. (1986). The natural variability of vital rates and associated statistics (with discussion). Biometrics 42 693--734.
  • Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • Gilbert, P. B., Lele, S. R. and Vardi, Y. (1999). Maximum likelihood estimation in semiparametric selection bias models with application to AIDS vaccine trials. Biometrika 86 27--43.
  • Gill, R. D., Vardi, Y. and Wellner, J. A. (1988). Large sample theory of empirical distributions in biased sampling models. Ann. Statist. 16 1069--1112.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel.
  • Huang, Y. and Wang, M.-C. (1995). Estimating the occurrence rate for prevalent survival data in competing risks models. J. Amer. Statist. Assoc. 90 1406--1415.
  • Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data, 2nd ed. Wiley, New York.
  • Keiding, N. (1990). Statistical inference in the Lexis diagram. Philos. Trans. Roy. Soc. London Ser. A 332 487--509.
  • Keiding, N., Kvist, K., Hartvig, H., Tvede, M. and Juul, S. (2002). Estimating time to pregnancy from current durations in a cross-sectional sample. Biostatistics 3 565--578.
  • Lexis, W. (1875). Einleitung in die Theorie der Bevölkerungsstatistik. Trübner, Strassburg. Pages 5--7 translated in (1977). Mathematical Demography (D. Smith and N. Keyfitz, eds.) 39--41. Springer, Berlin.
  • Lund, J. (2000). Sampling bias in population studies---How to use the Lexis diagram. Scand. J. Statist. 27 589--604.
  • Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York.
  • van Es, B., Klaassen, C. A. J. and Oudshoorn, K. (2000). Survival analysis under cross-sectional sampling: Length bias and multiplicative censoring. J. Statist. Plann. Inference 91 295--312.
  • Vardi, Y. (1982). Nonparametric estimation in the presence of length bias. Ann. Statist. 10 616--620.
  • Vardi, Y. (1985). Empirical distributions in selection bias models (with discussion). Ann. Statist. 13 178--205.
  • Vardi, Y. (1989). Multiplicative censoring, renewal processes, deconvolution and decreasing density: Nonparametric estimation. Biometrika 76 751--761.
  • Vardi, Y. and Zhang, C.-H. (1992). Large sample study of empirical distributions in a random-multiplicative censoring model. Ann. Statist. 20 1022--1039.
  • Wang, M.-C. (1991). Nonparametric estimation from cross-sectional survival data. J. Amer. Statist. Assoc. 86 130--143.
  • Wang, M.-C., Brookmeyer, R. and Jewell, N. P. (1993). Statistical models for prevalent cohort data. Biometrics 49 1--11.
  • Wang, M.-C., Jewell, N. P. and Tsai, W.-Y. (1986). Asymptotic properties of the product limit estimate under random truncation. Ann. Statist. 14 1597--1605.
  • Wolfson, C., Wolfson, D., Asgharian, M., M'Lan, C. E., Østbye, T., Rockwood, K. and Hogan, D., for the Clinical Progression of Dementia Study Group (2001). A reevaluation of the duration of survival after the onset of dementia. New England J. Medicine 344 1111--1116.