This paper focuses on the problem of the estimation of the cumulative hazard function of a distribution on d-dimensional Euclidean space when the data points are subject to censoring by an arbitrary adapted random set. A problem involving observability of the estimator proposed in [8] and [9] is resolved and a functional central limit theorem is proven for the revised estimator. Several examples and applications are discussed, and the validity of bootstrap methods is established in each case.
References
[1] Andersen, P.K., Borgan, O., Gill, R.D. and Keiding, N., Statistical Models Based on Counting Processes, Springer Series in Statistics, Springer-Verlag, New York, 1993.
[2] Carabarin Aguirre, A., Set-Indexed Survival Analysis with Generalized Censoring, doctoral thesis, University of Ottawa, 2008.
[3] David-Zadeh, O., Multi-Parameter Survival Analysis, doctoral thesis, Bar-Ilan University, 2008.
[4] Hildebrandt, T.H., Introduction to the Theory of Integration, Academic Press Inc., New York, 1963.
[5] Hougaard, Philip, Analysis of Multivariate Survival Data, Springer, New York, 2000.
[6] Ivanoff, B.G. and Merzbach, E., Set-Indexed Martingales, Chapman and Hall/CRC Press, Boca Raton, 2000.
[7] Ivanoff, B.G. and Merzbach, E., Set-Indexed Markov Processes, Canadian Mathematical Society Conference Proceedings 26, 2000, 217–232.
[8] Ivanoff, B.G. and Merzbach, E., Random censoring in set-indexed survival analysis, The Annals of Applied Probability 12, 2002, 944–971.
[9] Ivanoff, B.G. and Merzbach, E., Random clouds and an application to censoring in survival analysis, Stochastic Processes and their Applications 111, 2004, 259–279.
[10] Kalbfleisch, J.D. and Prentice, R.L. The Statistical Analysis of Failure Time Data, Wiley, New Jersey, 2002.
[11] Pons, O., A test of independence between two censored survival times, Scand. J. Statist. 13, 173–185 (1986).
[12] Pons, O. and Turckheim, E. de, Tests of independence for bivariate censored data based on the empirical joint hazard function, Scand. J. Statist. 18, 21–37 (1989).
[13] van der Vaart, Aad W. and Wellner, Jon A., Weak Convergence and Empirical Processes, Springer series in statistics, 1996.
