Multivariate calibration: A generalization of the classical estimator

In univariate calibration problems two different estimators are commonly in use. They are referred to as the classical estimator and the inverse estimator. [8], Technometrics 9, No. 3 425-439) compared these two methods of calibration by means of an extensive Monte Carlo study. Without mathematical proof he concluded that the classical estimator has a uniformly greater mean squared error than the inverse estimator. Krutchkoffs paper resulted in an immediate controversy on the subject of his criterion, for the classical estimator has an infinite mean and mean squared error. In this paper we consider a generalization of the classical estimator for multivariate regression problems. We show that this estimator has a finite mean if the dimension, say p, of the response variable is greater than 2, and we show that the mean squared error is finite if p is greater than 4. We also give exact expressions for the mean and the mean squared error in terms of expectations of Poisson variables, which can be easily approximated.