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

Abstract

We consider a failure hazard function, conditional on a time-independent covariate Z, given by ηgamma;0(t)fβ0(Z). The baseline hazard function ηγ0 and the relative risk fβ0 both belong to parametric families with θ0 = (β0,γ0)T ∈ ℝm+p. 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 fε. We observe a n-sample (Xi,Di,Ui), i = 1,..., n, where X i is the minimum between the failure time and the censoring time, and Di, is the censoring indicator. Using least square criterion and deconvolution methods, we propose a consistent estimator of θ0 using the observations (Xi,Di,Ui), i = 1,...,n. We give an upper bound for its risk which depends on the smoothness properties of fε and fβ(z) as a function of z, and we derive sufficient conditions for the √-consistency. We give detailed examples considering various type of relative risks fβ and various types of error density fε. In particular, in the Cox model and in the excess risk model, the estimator of θ0 is √n-consistent and asymptotically Gaussian regardless of the form of ε. © 2009 EDP Sciences SMAI.