Abstract :
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+
