Confidence regions in Cox proportional hazards model with measurement errors and unbounded parameter set

Abstract

Cox proportional hazards model with measurement errors is considered. In Kukush and Chernova (2017), we elaborated a simultaneous estimator of the baseline hazard rate $\lambda(\cdot)$ and the regression parameter $\beta$, with the unbounded parameter set $\varTheta=\varTheta_{\lambda}\times\varTheta_{\beta}$, where $\varTheta_{\lambda}$ is a closed convex subset of $C[0,\tau]$ and $\varTheta_{\beta}$ is a compact set in $\mathbb{R}^m$. The estimator is consistent and asymptotically normal. In the present paper, we construct confidence intervals for integral functionals of $\lambda(\cdot)$ and a confidence region for $\beta$ under restrictions on the error distribution. In particular, we handle the following cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.