Semi-parametric estimation of the hazard function in a model with covariate measurement error

We consider a model where the failure hazard function, conditional on a covariate $Z$ is given by $R(t,theta^0|Z)=eta_{gamma^0}(t)f_{beta^0}(Z)$, with $theta^0=(beta^0,gamma^0)^topin mathbb{R}^{m+p}$. The baseline hazard function $eta_{gamma^0}$ and relative risk $f_{beta^0}$ belong both to parametric families. The covariate $Z$ is measured through the error model $U=Z+epsilon$ where $epsilon$ is independent from $Z$, with known density $f_epsilon$. 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. We aim at estimating $theta^0$ in presence of the unknown density $g$. Our estimation procedure based on least squares criterion provide two estimators. The first one minimizes an estimation of the least squares criterion where $g$ is estimated by density deconvolution. Its rate depends on the smoothnesses of $f_epsilon$ and $f_beta(z)$ as a function of $z$,. We derive sufficient conditions that ensure the $sqrt{n}$-consistency. The second estimator is constructed under conditions ensuring that the least squares criterion can be directly estimated with the parametric rate. These estimators, deeply studied through examples are in particular $sqrt{n}$-consistent and asymptotically Gaussian in the Cox model and in the excess risk model, whatever is $f_epsilon$.