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

TL;DR

This paper develops methods to construct confidence regions for parameters in a Cox proportional hazards model with measurement errors, extending previous work to handle various error distributions and unbounded parameter sets.

Contribution

It introduces confidence interval and region construction techniques for the Cox model with measurement errors, accommodating unbounded parameters and different error distributions.

Findings

Confidence intervals for integral functionals of the baseline hazard rate.

Confidence regions for regression parameters.

Applicability to bounded, normal, and shifted Poisson error distributions.

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.