Nelson–Aalen and product-limit estimation in selection bias models for censored populations

Abstract

In this work we introduce a new model for inference when the data are sampled in a biased way from a censored population. This means that the risk of censoring occurs first, and afterwards some biased sampling takes place. Nelson–Aalen and product-limit type estimation is proposed, and strong iid representations of these estimates are provided. The introduced empirical distribution generalizes the time-honoured Kaplan–Meier product-limit to the censored-biased scenario.

Keywords:

• Biased sampling
• Censoring
• Kaplan–Meier
• Selection bias
• VC-class
• Weighted distributions

Acknowledgements

Thanks to a referee for comments and suggestions which have improved the presentation of the paper. The author is very grateful to Gema Álvarez Llorente for providing the Galician unemployment data. Work supported by the Grants PGIDIT02PXIA30003PR and BFM2002-03213.