LIKELIHOOD INFERENCE IN THE ERRORS-IN-VARIABLES MODEL

We consider estimation and confidence regions for the parameters alpha and beta based on the observations (X(1), Y-1),..., (X(n), Y-n,) in t he errors-in-variables model X(i) = Z(i) + e(i) and Y-i=alpha + beta Z (i)+ f(i) for normal errors e(i) and f(i) of which the covariance matr ix is known up to a constant. We study the asymptotic performance of t he estimators defined as the maximum likelihood estimator under the as sumption that Z(1),..., Z(n), is a random sample from a completely unk nown distribution. These estimators are shown to be asymptotically eff icient in the semi-parametric sense if this assumption is valid. These estimators are shown to be asymptotically normal even in the case tha t Z(1),Z(2),... are arbitrary constants satisfying a moment condition. Similarly we study the confidence regions obtained From the likelihoo d ratio statistic for the mixture model and show that these are asympt otically consistent both in the mixture case and in the case that Z(1) ,Z(2),... are arbitrary constants. (C) 1996 Academic Press, Inc.