Regression Diagnostics
When linear regression methods are applied to given data, useful results can be expected when the chosen model is considerably plausible, meaning that no substantial indications for inconsistencies and violation of model assumptions can be found. This con
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Lecture Notes in Statistics Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg, I. Olkin, N. Wermuth, S. Zeger
175
Jiirgen GraB
Linear Regression
Springer
Jiirgen GroB University of Dortmund Department of Statistics Vogelpothsweg 87 44221 Dortmund Germany [email protected]
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche N ationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-540-40178-0
ISBN 978-3-642-55864-1 (eBook)
DOI 10.1007/978-3-642-55864-1
This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerne O. Clearly it would be most helpful to have risk inequ aliti es between est imators for {3 being valid for all possible param eter values {3 , bu t t his will rarely t urn out to be the case if we consider reasonabl e est imators .
1 Fundamentals
12
Some Questions As we have demonstrat ed above , t here are sit ua t ions in which certain alt ernatives to the least squares est imator deliver smaller observed losses t han j!J. This, however , does not impl y t hat expected losses must behave similarly, meaning that we cannot dr aw any reasonabl e conclusions a bout t he behavi or of t he respective est imat ors from th e a bove example. Noneth eless, t he ab ove results raise some questions: - Is it possible th at a better est imator t han j!J exists for all possible f3 E lRP and all a 2 > O? In ot her words, does t here exist an esti mator which makes j!J in admissible for esti mating f3? - Do there exist different possibilities to define 'bet ter' ? Are there different reason able losses and risks? - Do there exist est imator s of f3 which are bet ter than j!J for cert ain sets of f3 E lRP and a 2 > O? In ot her words , do th ere exist est ima t ors which turn out to be adm issible compared to j!J? - If t here existed such admissible est imators , und er what conditio ns should t hey be used? - Ca n we find esti mators which are admissible compared to any ot her est imator? In such a case, we can never find a un iformly better est imator, i.e, an estimat or which is better for all f3 E lRP and all a 2 > 0 Terms like 'loss', 'risk' or 'admissibility' are widely used in decision theory. Therefore, before t ry ing to give some answers to the above questions, we pr esent a short int ro du ction into t his t heory.
1.2 Decision Theory and Point Estimation St atisti cal decision t heory has been established by Abra ham Wald with a series of pap ers in the 1940s being combined 1950 in his famous book St atistical Decision Functions [124]. Fur th er cont rib ut ions on t his to pic are for examp le provided by Ferguson [38] and Berger [13] . 1.2.1 Decision Rule Suppose we have to reach a decision d depending on p unknown quantities B1 , • . . , Bp • These quan ti t ies are combined in t he par am et
