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

Marie-Laure Martin-Magniette, Marie-Luce Taupin
2009 E S A I M: Probability & Statistics  
We consider a failure hazard function, conditional on a time-independent covariate Z, given by η γ 0 (t)f β 0 (Z). The baseline hazard function η γ 0 and the relative risk f β 0 both belong to parametric families with θ 0 = (β 0 , γ 0 ) ∈ R m+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 Xi is
more » ... 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 0 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 f β (z) as a function of z, and we derive sucient conditions for the √ n-consistency. We give detailed examples considering various type of relative risks f β and various types of error density fε. In particular, in the Cox model and in the excess risk model, the estimator of θ 0 is √ n-consistent and asymptotically Gaussian regardless of the form of fε. 1991 Mathematics Subject Classication. 62G05, 62F12,62G99, 62J02. The dates will be set by the publisher.
doi:10.1051/ps:2008004 fatcat:nrtkb4cm6zcsnahapdzdn6m76i