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Nonparametric estimation for censored lifetimes suffering from unknown selection bias

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Abstract

In a population of individuals, where the random variable (r.v.) σ denotes the birth time and X the lifetime, we consider the case, where an individual can be observed only if its life-line \(\mathcal{L}\)(σ, X) = {(σ + y, y), 0 ≤ y ≤ X} intersects a given Borel set S in ℝ × ℝ+. Denoting by σ S and X S the birth time and lifetime for the observed individuals, we point out that the distribution function (d.f.) F S of the r.v. X S suffers from a selection bias in the sense that F S = ∝ w d F/μ S, where w and μ S depend only on the distribution of σ and on F, the d.f. of X. Assuming in addition that the r.v. X S is randomly right-censored as soon as the individual is selected, we construct a productlimit estimator \(\hat F_\mathcal{S} \) for the d.f. F S and a nonparametric estimator ŵ for the weight function w. We prove a consistency result for ŵ and a weak convergence result for \(\hat F_\mathcal{S} \). We establish in addition an exponential bound for \(\hat F_\mathcal{S} \).

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References

  1. P.K. Andersen, O. Borgan, R. D. Gill, and N. Keiding, Statistical Models based on Counting Processes (Springer, 1993).

  2. M. Asgharian, C. E. M’Lan, and D. B. Wolfson, “Length-biased sampling with right censoring: an unconditional approach”, J. Amer. Statist. Assoc. 97, 201–209 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Asgharian, “Biased sampling with right censoring: a note on Sun, Cui and Tiwari”, (2002). Canadian J. Statist. 30, 475–490 (2003).

    Article  Google Scholar 

  4. D. Bitouzé, B. Laurent, and P. Massart, “A Dvoretsky-Kiefer-Wolfowitz type inequality for the Kaplan—Meier estimator”, Ann. Inst. H. Poincaré 32, 735–763 (1999).

    Article  Google Scholar 

  5. K. Chen, M.-T. Chao, and S.-H. Lo, “On strong uniform consistency of the Lynden—Bell estimator for truncated data”, Ann. Statist. 23, 440–449 (1995).

    MATH  MathSciNet  Google Scholar 

  6. S. Efromovich, “Distribution estimation for biased data”, J. Statist. Plann. Inference 124, 1–43 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  7. R. D. Gill, “Large sample behaviour of the product-limit estimator on the whole line”, Ann. Statist. 11, 49–58 (1983).

    MATH  MathSciNet  Google Scholar 

  8. R. D. Gill, Y. Vardi, and J. A. Wellner, “Large sample theory of empirical distributions in biased sampling models”, Ann. Statist. 16, 1069–1172 (1988).

    MATH  MathSciNet  Google Scholar 

  9. R. D. Gill and N. Keiding, “Random truncation models and Markov processes”, Ann. Statist. 18, 582–602 (1990).

    MATH  MathSciNet  Google Scholar 

  10. N. Keiding, “Statistical inference for the Lexis diagram”, Roy. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 332, 487–509 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  11. N. Keiding, “Age-specific incidence and prevalence: a statistical perspective. With discussion”, J. Roy. Statist. Soc. Ser. A 154, 371–412 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  12. N. Keiding, “Event history analysis and the cross-section”, Statist. Med. 25, 2343–2364 (2006).

    Article  MathSciNet  Google Scholar 

  13. J. F. C. Kingman, Poisson Processes (Oxford Science Publ., 1993).

  14. E. Lenglart, “Relation de domination entre deux processus”, Ann. Inst. H. Poincaré 13, 171–179 (1977).

    MathSciNet  Google Scholar 

  15. W. Lexis, (1875). “Einleitung in die Theorie der Bevölkerung-Statistik”, in Mathematical Demography, Ed. by D. Smith and N. Keyfitz (Springer, Berlin, 1875); Biomathematics 6, 39–41 (1977).

    Google Scholar 

  16. J. Lund, “Sampling bias in population studies. How to use the Lexis diagram”, Scand. J. Statist. 27, 589–604 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Rebolledo, “Central limit theorems for local martingales”, Z. Wahrsch. verw. Gebiete. 51, 269–286 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  18. J. de Uña-Àlvarez, “Product-limit estimation for length-biased censored data”, Test 11, 109–125 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  19. J. de Uña-Àlvarez, “Nonparametric estimation under length-biased sampling and type I censoring: a moment based approach”, Ann. Inst. Statist. Math. 56, 667–681 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  20. J. de Uña-Àlvarez and Á. Saavedra, “Bias and variance of the nonparametric MLE under length-biased censored sampling: a simulation study”, Comm. Statist. Simulation Comput. 33, 397–413 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  21. J. de Uña-Àlvarez, “Nelson-Aalen and product-limit estimation in selection bias models for censored populations”, J. Nonparam. Statist. 16, 761–777 (2004).

    Article  MATH  Google Scholar 

  22. S. van de Geer, “Hellinger consistency of certain nonparametric maximum likelihood estimators”, Ann. Statist. 21, 14–44 (1993).

    MATH  MathSciNet  Google Scholar 

  23. M. J. Van der Laan, Proving Efficiency of NPMLE and Identities, Technical Report no. 44, PhD thesis (Dept. of Math., Univ. of Utrecht, 1994).

  24. Y. Vardi, “Nonparametric estimation in presence of length bias”, Ann. Statist. 10, 616–620 (1982).

    MATH  MathSciNet  Google Scholar 

  25. M.-C. Wang, N. P. Jewell, and W. Y. Tsai, “Asymptotic properties of the product limit estimate under random truncation”, Ann. Statist. 14, 1597–1605 (1986).

    MATH  MathSciNet  Google Scholar 

  26. B. B. Winter and A. Földes, “A product-limit estimator for use with length-biased data”, Canad. J. Statist., 16, 337–355 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Woodroofe, “Estimating a distribution function with truncated data”, Ann. Statist. 13, 163–177 (1985).

    MATH  MathSciNet  Google Scholar 

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  1. LSTA, Universite Pierre et Marie Curie — Paris 6, 175 rue du Chevaleret, 75013, Paris, France

    A. Guilloux

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  1. A. Guilloux
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Guilloux, A. Nonparametric estimation for censored lifetimes suffering from unknown selection bias. Math. Meth. Stat. 16, 202–216 (2007). https://doi.org/10.3103/S1066530707030027

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  • DOI: https://doi.org/10.3103/S1066530707030027

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