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

Abstract

Cox proportional hazards model with measurement error is investigated. In Kukush et al. (2011) [Journal of Statistical Research 45, 77-94] and Chimisov and Kukush (2014) [Modern Stochastics: Theory and Applications 1, 13-32] asymptotic properties of simultaneous estimator ${\lambda }_n(\cdot )$, ${\beta }_n$ were studied for baseline hazard rate $\lambda (\cdot )$ and regression parameter $\beta$, at that the parameter set $\Theta ={\Theta }_{\lambda }\times {\Theta }_{\beta }$ was assumed bounded. In the present paper, the set ${\Theta }_{\lambda }$ is unbounded from above and not separated away from $0$. We construct the estimator in two steps: first we derive a strongly consistent estimator and then modify it to provide its asymptotic normality.