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Biometrika Advance Access published online on October 22, 2008
Biometrika, doi:10.1093/biomet/asn044
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© 2008 Biometrika Trust

Article

A note on nonparametric quantile inference for competing risks and more complex multistate models

Jan Beyersmann

Freiburg Centre for Data Analysis and Modelling, University of Freiburg, 79104 Freiburg, Germany jan.beyersmann{at}fdm.uni-freiburg.de

Martin Schumacher

Institute of Medical Biometry and Medical Informatics, University Medical Centre Freiburg, 79104 Freiburg, Germany ms{at}imbi.uni-freiburg.de

Received for publication 1 October 2007. Revision received 1 April 2008.
  Abstract

Nonparametric quantile inference for competing risks has recently been studied by Peng & Fine (2007). Their key result establishes uniform consistency and weak convergence of the inverse of the Aalen–Johansen estimator of the cumulative incidence function, using the representation of the cumulative incidence estimator as a sum of independent and identically distributed random variables. The limit process is of a form similar to that of the standard survival result, but with the cause-specific hazard of interest replacing the all-causes hazard. We show that this fact is not a coincidence, but can be derived from a general Hadamard differentiation result. We discuss a simplified proof and extensions of the approach to more complex multistate models. As a further consequence, we find that the bootstrap works.

Key Words: Cumulative incidence function • Functional delta method • Inverse functional • Survival analysis


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