Estimators for the errors-in-variables model

Estimators of the parameters of the multivariate linear errors-in-variables model and the nonlinear errors-in-variables model are investigated. The multivariate linear errors-in-variables model is defined by; Y(,t) = (beta)(,0) + x(,t)(beta) + e(,t),; X(,t) = x(,t) + u(,t), t = 1,2,...,n,;where Y(,t) and X(,t) are observable random row vectors of dimensions r and k, respectively, e(,t) and u(,t) are unobservable error vectors, x(,t) is an unobservable random or fixed vector, (beta)(,0) is a 1 x r vector of parameters, (beta) is a k x r matrix of parameters, and (epsilon)(,t) = (e(,t), u(,t)) are independently and identically distributed with mean zero and covariance matrix (SIGMA)(,(epsilon)(epsilon)). It is assumed that an independent estimator S(,(epsilon)(epsilon)) of (SIGMA)(,(epsilon)(epsilon)) is available;Under the assumption that the (epsilon)(,t) are normally distributed and that S(,(epsilon)(epsilon)) is a multiple of a Wishart matrix, the maximum likelihood estimators are obtained for the model with fixed x(,t) and the model with normally distributed x(,t). The asymptotic properties of the estimators are derived under minimal assumptions;The nonlinear errors-in-variables model is defined by; Y(,nt) = f(x(,t); (beta)) + e(,nt),; X(,nt) = x(,t) + u(,nt), t = 1,2,...,b(,n),;where x(,t) is an unobservable fixed row vector of dimension q, (Y(,nt), X(,nt)) are observed in the n-th experiment, (beta) is a k x 1 vector of parameters, and (e(,nt), u(,nt)) are independently distributed with mean zero and covariance matrix (SIGMA)(,n). It is assumed that n = a(,n)b(,n) for all n and that;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);For the nonlinear model with known (SIGMA)(,n), the asymptotic bias of the normal maximum likelihood estimator is obtained. A class of estimators adjusted for the nonlinearity bias is given. Three estimators in the class are discussed and compared in a Monte Carlo study;The instrumental variable estimator (\u27)(beta)(,n) of (beta) is defined for the nonlinear model with unknown (SIGMA)(,n) when additional information is provided by an observable vector W(,nt). The asymptotic properties of (\u27)(beta)(,n) and of estimators of (SIGMA)(,n) and x(,t) based on (\u27)(beta)(,n) are derived. A modified instrumental variable estimator (\u27)(beta)(,n) of (beta) is constructed using (\u27)(beta)(,n) as a preliminary estimator. The estimator (\u27)(beta)(,n) is shown to be asymptotically more efficient than (\u27)(beta)(,n)