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 $f_\varepsilon$
.