Error Matrix Criterion Research Articles

Pre-test estimation in the linear regression model with competing restrictions

Consider the linear regression model M={ y, Xβ, σ 2 I n } and two sets of competing, not necessarily exact linear restrictions R j> β= r j , j=1, 2. Assume that the linear restrictions are nested, that is, R 1= T R 2 and r 1= Tr 2 for some matrix T. To estimate the vector β, we derive the pre-test estimator b̄, a two stage procedure based on testing the dominance condition which follows the mean squared error matrix comparison of b 1 and b 2, the restricted least squares estimators of β in the model M j ={ y, Xβ| R j = r j , σ 2 I n }, j=1, 2. We investigate statistical properties of the pre-test estimator b̄ and, comparing b̄ with b 1, characterize its optimality under the mean squared error matrix criterion.

Mean square error matrix improvements and admissibility of linear estimators

Abstract In the first part of this paper, the set L (Cy + c), comprising all linear estimators of β which are as good as a given unbiased estimator Cy + c with respect to the mean square error matrix criterion in at least one point of the parameter space, is investigated under the unrestricted linear regression model M = {y, Xβ, σ2In} and the restricted model M0 = {y, Xβ¦R0β = r0, σ2/In}. In the second part, new characterizations of the sets A and A 0 of all linear estimators that are admissible for β under M and M0 with respect to the mean square error criterion are derived via appropriately reducing the sets L (β^) and L (β^0), where β^ and β^0 are the minimum dispersion linear unbiased estimators of β in these two models. The convexity of the sets L (Cy + c), A , and A 0 is also pointed out.

Mean square error matrix superiority of estimators under linear restrictions and misspecification

In this paper attention is focussed on the mixed regression estimator (MRE), which is based upon incorrect prior information, and on the restricted least squares estimator (RLSE), which uses valid or invalid restrictions. With respect to the mean square error (MSE) matrix criterion we can derive conditions, under which MRE and RLSE are better than the ordinary least squares estimator (OLSE), especially in the case that the regression model is misspecified. Finally we examine the superiority of the MSE-predictor over the OLS-predictor.

A matrix inequality and admissibility of linear estimators with respect to the mean square error matrix criterion

Given matrices A, B and vectors a, b, a necessary and sufficient condition is established for the Löwner partial ordering ( Am+ a)( Am+ a)′ ⩽ ( Bm+ b)( Bm+ b)′ to hold for all vectors m. This result is then applied to derive a complete characterization of estimators that are admissible for a given vector of parametric functions among the set of all linear estimators under the general Gauss-Markov model, when the mean square error matrix is adopted as the criterion for evaluating the estimators.

Mean square error matrix superiority of the mixed regression estimator under misspecification

Conditions are derived under which the mixed regression estimator (MRE) is better then the ordinary least-squares estimator (OLSE) with respect to the mean square error (MSE) matrix criterion especially for the case that the regression model is misspecified. Some attention is paid to prediction, where it is shown that the MRE-predictor is potentially superior to the OLS-predictor under the same criterion.