Empirical likelihood confidence bands for mean functions of recurrent events with competing risks and a terminal event

¹ Universitéde Toulouse – INSA, IMT UMR CNRS 5219, 135 Avenue de Rangueil, 31077 Toulouse cedex 4, France.
² Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comte, 16 route de Gray, 25030 Besançon cedex, France.
davit.varron@gmail.com

Received: 17 September 2014
Revised: 8 February 2016
Accepted: 19 February 2016

Abstract

In this paper, we consider recurrent events with competing risks in the presence of a terminal event and a censorship. We focus our attention on the mean functions which give the expected number of events of a specific type that have occurred up to a time t. Using heuristics from empirical likelihood theory, we propose a method to build simultaneous (in t) confidence regions for these functions. To establish the consistency of this estimation method (as well as its bootstrap calibration), we prove a weak convergence (as stochastic processes) of the associated empirical likelihood ratio processes. Our approach almost entirely relies on empirical process methods. In the proofs, we also establish some results in empirical processes theory that may present some independent interest. Then we carry out a simulation study of our confidence bands, we compare those obtained by empirical likelihood to the ones obtained by bootstrap. Finally, our procedure is applied on a real data set of nosocomial infections in an intensive care unit of a French hospital.