This paper studies semiparametric maximum likelihood estimators in the Cox proportional hazards model with covariate error, assuming that the conditional distribution of the true covariate given the surrogate is known. We show that the estimator of the regression coefficient is asymptotically normal and efficient, its covariance matrix can be estimated consistently by differentiation of the profile likelihood, and the likelihood ratio test is asymptotically chi-squared. We also provide efficient algorithms for the computations of the semiparametric maximum likelihood estimate and the profile likelihood. The performance of this method is successfully demonstrated in simulation studies.
