A consistent estimator in the accelerated failure time model with censored observations and measurement errors

We consider the following accelerated failure time model used in the statistical analysis of the survival data: \[ T_i=\exp \bigl \{\beta _0+\beta _X^T X_i+\varepsilon _i\bigr \}, \qquad i\geq 1.\] The lifetimes $T_i$ are observed under censoring. We also observe the vectors $W_i=X_i+U_i$ instead of the regressors $X_i$, where the $U_i$ are measurement errors. The vector of regression parameters $\beta =\bigl (\beta _0,\beta _X^T\bigr )^T$ is estimated from the observations. We construct an estimator as a solution of the corresponding unbiased estimating equation and show that this estimator is consistent if the censoring distribution is known. We also prove the consistency of the estimators for the case of an unknown censoring distribution if the regressors $X_i$ are bounded and the errors $\varepsilon _i$ are bounded from above. For the latter case, we estimate the censoring distribution by the Kaplan–Meier method.