Likelihood Inference in the Errors-in-Variables Model

We consider estimation and confidence regions for the parametersαandβbased on the observations (X1, Y1), …, (Xn, Yn) in the errors-in-variables modelXi=Zi+eiandYi=α+βZi+fifor normal errorseiandfiof which the covariance matrix is known up to a constant. We study the asymptotic performance of the estimators defined as the maximum likelihood estimator under the assumption thatZ1, …, Znis a random sample from a completely unknown distribution. These estimators are shown to be asymptotically efficient in the semi-parametric sense if this assumption is valid. These estimators are shown to be asymptotically normal even in the case thatZ1, Z2, … are arbitrary constants satisfying a moment condition. Similarly we study the confidence regions obtained from the likelihood ratio statistic for the mixture model and show that these are asymptotically consistent both in the mixture case and in the case thatZ1, Z2, … are arbitrary constants.