Journal of Probability and Statistics
Volume 2011 (2011), Article ID 182049, 15 pages
Research Article

Determinant Efficiencies in Ill-Conditioned Models

Department of Statistics, Virginia Polytechnic Institute, Blacksburg, VA 24061, USA

Received 18 May 2011; Accepted 1 August 2011

Academic Editor: Michael Lavine

Copyright © 2011 D. R. Jensen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The canonical correlations between subsets of OLS estimators are identified with design linkage parameters between their regressors. Known collinearity indices are extended to encompass angles between each regressor vector and remaining vectors. One such angle quantifies the collinearity of regressors with the intercept, of concern in the corruption of all estimates due to ill-conditioning. Matrix identities factorize a determinant in terms of principal subdeterminants and the canonical Vector Alienation Coefficients between subset estimators—by duality, the Alienation Coefficients between subsets of regressors. These identities figure in the study of D and 𝐷 𝑠 as determinant efficiencies for estimators and their subsets, specifically, 𝐷 𝑠 -efficiencies for the constant, linear, pure quadratic, and interactive coefficients in eight known small second-order designs. Studies on D- and 𝐷 𝑠 -efficiencies confirm that designs are seldom efficient for both. Determinant identities demonstrate the propensity for 𝐷 𝑠 -inefficient subsets to be masked through near collinearities in overall D-efficient designs.