A linear empirical Bayes solution for the calibration problem

Abstract

The well-known problem of multivariate calibration involves making inferences about an unknown q×1 vector X from a single random observed p×1 response vector Y. The relationship between Y and X is calibrated with experimental data ( X i , Y i ), i=1,2,…, n, where Y i and X i are p×1 and q×1 vectors, respectively, and satisfy the regression relationship Y i = α+ β′ X i + ε i , with ε i being the random error, and α( p×1) and β( q× p) are parameters. Two well-known competing estimators of X are the classical and inverse estimators. They can be obtained by direct or inverse regression and are supported by the maximum likelihood and Bayesian approaches, respectively. In this paper we exhibit another estimator, which is a convex combination of n known values of X i, (i=1,…,n) and the classical estimator of the unknown X 0, derived by exploiting the techniques of linear empirical Bayes methods. We demonstrate that the proposed estimator has better performance over the classical estimator in the sense of having a smaller generalized mean squared error. We also compare our estimator with the inverse estimator as well. A simulation study is used to help validate the performance of the proposed estimator in small samples. Three practical examples are also analyzed. Our results show that the proposed estimator is overall the best choice among the three.