Estimation in a competing risks proportional hazards model under length-biased sampling with censoring |
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Authors: | Jean-Yves Dauxois Agathe Guilloux Syed N U A Kirmani |
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Institution: | 1. Université de Toulouse-INSA, IMT, UMR CNRS 5219, 135, Avenue de Rangueil, 31077, Toulouse cedex 4, France 2. Laboratoire de Statistique Théorique et Appliquée (LSTA), équipe d’Accueil 3124, 4 place Jussieu, 75252, Paris cedex 05, France 3. Department of Mathematics, University of Northern Iowa, Cedar Falls, IA, 50614-0506, USA
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Abstract: | What population does the sample represent? The answer to this question is of crucial importance when estimating a survivor function in duration studies. As is well-known, in a stationary population, survival data obtained from a cross-sectional sample taken from the population at time $t_0$ represents not the target density $f(t)$ but its length-biased version proportional to $tf(t)$ , for $t>0$ . The problem of estimating survivor function from such length-biased samples becomes more complex, and interesting, in presence of competing risks and censoring. This paper lays out a sampling scheme related to a mixed Poisson process and develops nonparametric estimators of the survivor function of the target population assuming that the two independent competing risks have proportional hazards. Two cases are considered: with and without independent censoring before length biased sampling. In each case, the weak convergence of the process generated by the proposed estimator is proved. A well-known study of the duration in power for political leaders is used to illustrate our results. Finally, a simulation study is carried out in order to assess the finite sample behaviour of our estimators. |
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