The Coordinate-Free Approach to Gauss-Markov Estimation
These notes originate from a couple of lectures which were given in the Econometric Workshop of the Center for Operations Research and Econometrics (CORE) at the Catholic University of Louvain. The participants of the seminars were recommended to read the
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40 Hilmar Drygas Studiengruppe fUr Systemforschung, Heidelberg
The Coordinate-Free Approach to Gauss-Markov Estimation
Springer-Verlag Berlin· Heidelberg· New York 1970
Advisory Board H. Albach . A. v. Balakrishnan· F. Ferschl R. E. Kalman· W. Krelle . N. Wirth
ISBN-13: 978-3-540-05326-2 DOl: 10.1007/978-3-642-65148-9
e-ISBN-13: 978-3-642-65148-9
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the tee to be determined by agreement with the publisher. © by Springer-Verlag Berlin· Heidelberg 1970. Library of Congress Catalog Card Number 78-147405.
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Summary and Preface.
These notes originate from a couple of lectures which were given in the Econometric Workshop of the Center for Operations Research and Econometrics (CORE) at the Catholic University of Louvain.
The
participants of the seminars were recommended to read the first four chapters of Seber's book [40], but the exposition of the material went beyond Seber's exposition, if it seemed necessary. Coordinate-free methods are not new in Gauss-Markov estimation, besides Seber the work of Kolmogorov [11], SCheffe [36], Kruskal [21], [22] and Malinvaud [25], [26] should be mentioned.
Malinvaud's
approach however is a little different from that of the other authors, because his optimality criterion is based on the ellipsoid of concentration.
This criterion is however equivalent to the usual con-
cept of minimal covariance-matrix and therefore the result must be the same in both cases.
While the usual theory gives no indication
how small the covariance-matrix can be made before the optimal estimator is computed, Malinvaud can show how small the ellipsoid of concentration can be made:
it is at most equal to the intersection
of the ellipssoid of concentration of the observed random vector and the linear space in which the (unknown) expectation value of the observed random vector is lying. This exposition is based on the observation, that in regression ~nalysis
and related fields two conclusions are or should preferably
be applied repeatedly.
The first important fundamental lemma is the
Farkas' theorem, which is closely related to the well-known famous
IV
It is
Farkas-Minkowski theorem (see e.g. Gale [12], pp. 41-49).
mainly based on the definition of the adjoint mapping, or to express it in matrices, on the definition of the transposed matrix. Chipman [4] has already pointed out this close relationship.
The
second important lemma is the projection theorem, which says, that a given point outside of a linear manifold
has minimal distance
from a point on the linear manifold if and only if the connecting line between the two points
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