Consistent estimation in Cox proportional hazards model with measurement errors and unbounded parameter set

We study the Cox problem with proportional risks and measurement errors. The asymptotic properties of the simultaneous estimator $\lambda _n({\boldsymbol \cdot })$, $\beta _n$ of the baseline hazard function $\lambda ({\boldsymbol \cdot })$ and regression parameter $\beta$ are considered in the papers [6] and [3] for the case of a bounded set of parameters $\Theta =\Theta _{\lambda }\times \Theta _{\beta }$. In the current paper, the set $\Theta _{\lambda }$ is unbounded from above and is not separated from zero. The estimator is constructed in the following two steps. First, one obtains a strictly consistent estimator and, second, this estimator is corrected in order to obtain an asymptotically normal estimator.