ADAPTIVE ESTIMATORS OF A MEAN MATRIX: TOTAL LEAST SQUARES VERSUS TOTAL SHRINKAGE

An unknown constant matrix M is observed with additive random error+ The basic problem considered is to devise an estimator of M that trades off bias against variance so as to achieve relatively low quadratic risk+ This paper develops an adaptive total least squares estimator and an adaptive total shrinkage estimator of M that minimize estimated risk over certain large classes of linear estimators+ It is shown that the asymptotic risk of the adaptive total least squares estimator is the smallest attainable among reduced rank total least squares fits to the data matrix+ The asymptotic risk of the adaptive total shrinkage estimator is shown to be smaller still+ A close link is established between total shrinkage and the Efron–Morris estimator of M+ In the asymptotics, the row dimension of M tends to infinity, and the column dimension stays fixed+ The risks converge uniformly when the signalto- noise ratio and the measurement error variance are both bounded+ A second problem treated is estimation of M under the assumption that a linear relation holds among its columns+ In this formulation of the errors-in-variables linear regression model, rank constrained adaptive total least squares asymptotically dominates the usual total least squares estimator of M, and rank constrained adaptive total shrinkage is better still+