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

We consider a failure hazard function, conditional on a time-independent covariate Z, given by ηγ0(t)fβ0(Z). The baseline hazard function ηγ0 and the relative risk fβ0 both belong to parametric families with θ = (β, γ0)> ∈ R. 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 Xi 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 θ 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 su cient conditions for the √ n-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 θ is √ n-consistent and asymptotically Gaussian regardless of the form of fε. 1991 Mathematics Subject Classi cation. 62G05, 62F12,62G99, 62J02. The dates will be set by the publisher.