We consider a failure hazard function, conditional on a time-independent covariate Z, given by $\eta_{\gamma^0}(t)f_{\beta^0}(Z)$. The baseline hazard function $\eta_{\gamma^0}$ and the relative risk $f_{\beta^0}$ both belong to parametric families with $\theta^0=(\beta^0,\gamma^0)^\top\in \mathbb{R}^{m+p}$. 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 $f_\varepsilon$. 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 θ0 using the observations n-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its risk which depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ as a function of z, and we derive sufficient conditions for the $\sqrt{n}$-consistency. We give detailed examples considering various type of relative risks $f_{\beta}$ and various types of error density $f_\varepsilon$. In particular, in the Cox model and in the excess risk model, the estimator of θ0 is $\sqrt{n}$-consistent and asymptotically Gaussian regardless of the form of $f_\varepsilon$.