Deepest Minimum Criterion for Biased Affine Estimation

Abstract

A new strategy called the Deepest Minimum Criterion (DMC) is presented for optimally obtaining an affine transformation of a given unbiased estimator, when a-priori information on the parameters is known. Here, it is considered that the samples are drawn from a distribution parametrized by an unknown deterministic vector parameter. The a-priori information on the true parameter vector is available in the form of a known subset of the parameter space to which the true parameter vector belongs. A closed form exact solution is given for the non-linear DMC problem in which it is known that the true parameter vector belongs to an ellipsoidal ball and the covariance matrix of the unbiased estimator does not depend on the parameters. A closed form exact solution is also given for the Min-Max strategy for this same case.

Analysis and Comparison of Biased Affine Estimators

“…Usually, is taken to be the identity matrix and this will be assumed from now on as both the MM [5] and the LMI [6] estimators do not consider a weighted mean squared error. For the general case please refer to [7].…”

“…The matrix is, in general, a function of and . From (2), the mean squared error of the biased estimator is (3) From now on, it will be considered the practical case when , see [7]. This implies that there is room for some actual improvement of the biased affine estimator over the unbiased estimator.…”

“…Mainly, the Min-Max (MM) technique [5], the Linear Matrix Inequality (LMI) method [6] and the Deepest Minimum Criterion (DMC) strategy [7]. Each technique looks to obtain the affine transformation following different optimality criteria.…”

“…The MM strategy [5] is a conservative approach that tries to uniformly reduce the mean squared error while the LMI criterion [6] is a more strict strategy that lowers the mean squared error in every possible axis of the parameter space. The DMC [7] tries to lower the minimum value of the mean squared error of the biased estimator while ensuring that the affine estimator dominates over the unbiased estimator. Nonetheless, all three strategies yield admissible estima-tors.…”

“…The general problem of affine biased estimation for the case of constant covariance matrix and ellipsoidal constraints is considered in Section II (for a more general treatment on the subject, see [7]). Also, conclusions can be found in Section VIII.…”