Estimation of the hazard function in a semiparametric model with covariate measurement error

¹ UMR AgroParisTech/INRA MIA 518, Paris, France.
² URGV UMR INRA 1165/CNRS 8114/UEVE, Évry, France.
³ Université Paris Descartes, Laboratoire MAP5, Paris; marie-luce.taupin@math-info.univ-paris5.fr

Received: 2 April 2007
Revised: 30 January 2008

Abstract

We consider a failure hazard function, conditional on a time-independent covariate Z, given by . The baseline hazard function and the relative risk both belong to parametric families with . The covariate Z has an unknown density and is measured with an error through an additive error model U = Z + ε where ε is a random variable, independent from Z, with known density . We observe a n-sample (X_i, D_i, U_i), i = 1, ..., n, where X_i is the minimum between the failure time and the censoring time, and D_i is the censoring indicator. Using least square criterion and deconvolution methods, we propose a consistent estimator of θ⁰ using the observations n-sample (X_i, D_i, U_i), i = 1, ..., n.
We give an upper bound for its risk which depends on the smoothness properties of and as a function of z, and we derive sufficient conditions for the -consistency. We give detailed examples considering various type of relative risks and various types of error density . In particular, in the Cox model and in the excess risk model, the estimator of θ⁰ is -consistent and asymptotically Gaussian regardless of the form of .