Parameter Estimation and Hypothesis Testing in Linear Models PDF
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Parameter Estimation and Hypothesis Testing in Linear Models Springer-Verlag Berlin Heidelberg GmbH Karl-Rudolf Koch Parameter Estimation and Hypothesis Testing in Linear Models Second, updated and enlarged Edition With 20 Figures " Springer Prof. Dr.-Ing., Dr.-Ing. E. h. Karl-Rudolf Koch Institute of Theoretical Geodesy of the University of Bonn Nussallee 17 D-53115 Bonn Gennany This is a translation of the 3n1 German Edition "Parameterschiitzung und Hypothesentests in !inearen Modellen" © Ferd. Dummlers Verlag, 1997 ISBN 978-3-642-08461-4 CIP data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Koch, Karl-Rudolf: Parameter estimation and hypothesis testing in linear models I Karl-Rudolf Koch. -2., upd. and enl. ed. Einheitssacht.: Parameterschlltzung und Hypothesentests in Iinearen Modellen <eng!.> ISBN 978-3-642-08461-4 ISBN 978-3-662-03976-2 (eBook) DOI 10.1007/978-3-662-03976-2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Copyright Law. Violations are liable for prosecution act under German © Springer-Verlag Berlin Heidelberg 1999 Originally published by Springer-Verlag Berlin Heidelberg New Yark m1 999 Tht use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by author Cover design: E. Kirchner, Heidelberg SPIN: 10696502 3213020 - 5 43 2 1 0 - Printed on acid -free paper Preface to the Second Edition The necessity to publish the second edition of this book arose when its third German edition had just been published. This second English edition is there fore a translation of the third German edition of Parameter Estimation and Hypothesis Testing in Linear Models, published in 1997. It differs from the first English edition by the addition of a new chapter on robust estimation of parameters and the deletion of the section on discriminant analysis, which has been more completely dealt with by the author in the book Bayesian In ference with Geodetic Applications, Springer-Verlag, Berlin Heidelberg New York, 1990. Smaller additions and deletions have been incorporated, to im prove the text, to point out new developments or to eliminate errors which became apparent. A few examples have been also added. I thank Springer-Verlag for publishing this second edition and for the assistance in checking the translation, although the responsibility of errors remains with the author. I also want to express my thanks to Mrs. Ingrid Wahl and to Mrs. Heidemarlen Westhiiuser who prepared the second edition. Bonn, January 1999 Karl-Rudolf Koch Preface to the First Edition This book is a translation with slight modifications and additions of the second German edition of Parameter Estimation and Hypothesis Testing in Linear Models, published in 1987. As already mentioned in the preface to the first and second German edition, the book intends to give a self-contained presentation of the multidimensional methods of estimating parameters, test ing hypotheses and estimating intervals. In addition to these topics, vector and matrix algebra as well as probability theory are therefore presented. To deal with all this material, a condensed writing style is necessary which, how ever, is interspersed with many examples. The book is addressed especially to engineers, to those still studying and to those applying statistical methods to practical problems. But also members of other professions should find advice in this book helpful when dealing with statistical inference. I am indebted to Springer-Verlag for publishing the English translation. Assistance of Springer-Verlag in checking the translation is gratefully ac knowledged, although the responsibility of errors remains with the author. I would like to thank colleagues and students at the Department of Survey ing Engineering of the University of Calgary for eliminating errors in the manuscript, which was used as the basis for a course in statistical inference in the autumn of 1987. I also would like to express my gratitude to Mrs. Karin Bauer who prepared the copy of the book. Bonn, December 1987 Karl-Rudolf Koch Preface to the Third (German) Edition For the third edition, which became necessary, this book has again been completely revised. A section on robust estimation of parameters has been added and the chapter on discriminant analysis has been deleted, since this topic was dealt with more completely by the author in the book Bayesian In ference with Geodetic Applications, Springer-Verlag, Berlin Heidelberg New York, 1990. Some additional examples have been included, to give a clearer idea of the presentation. Smaller additions or deletions have been incorpo rated to improve the text, to point out newer developments or to remove errors which became apparent. I want to thank all students of our university, who made suggestions to improve the book. I also express my thanks to the co-workers of the Institute of Theoretical Geodesy, who assisted in the new edition. My special thanks go to Mr. Robert Blinken, Dipl.-Ing., who made several suggestions for the new edition, and to Mrs. Ingrid Wahl for writing and designing the text and the figures of this third edition. Finally I would like to mention the good cooperation with the publisher. Bonn, March 1997 Karl-Rudolf Koch Preface to the Second (German) Edition The necessity to come out with a second edition of this book was used to revise it completely. Again, the endeavor was made to give a self-contained presentation of the methods of estimating unknown parameters, of testing hypotheses and of interval estimation and to add necessary knowledge on linear algebra and probability theory. The structure of the book therefore has not been changed. But several examples have been added, in order to loosen up the generally compact presentation of the methods. Where the text had to be improved or where new develoments had to be pointed out, passages have been added or deleted and errors, which became apparent, were removed. More extensive additions to be mentioned are the direct computation of the Cholesky factorization, the properties of a special symmetrical reflexive generalized inverse, the non-central Wishart distribution for the matrices of quadratic forms, the interpretation of the constraints to remove the rank deficiency in a free network, the datum transformation, the Gauss-Markoff model for outliers in the observations, the simplified iterated estimates of the variance and covariance components, the less sensitive tests of hypotheses in the univariate and multivariate model of estimating parameters, the compu tation of the probability of Type II errors for tests in the multivariate model, the confidence region for several linear functions of the unknown parameters, the derivation of the test for outliers as a special case of the test of a gen eral hypothesis in the univariate model, the measures of reliability and their interpretation. I want to thank all the colleagues and students, who have contributed to this edition by their suggestions for improvements. I also convey my thanks to my co-workers in the Institute of Theoretical Geodesy, who have assisted in the preparation of the new edition. In particular, I thank Mrs. Stefanie Schulte, Dipl.-Ing., for her suggestions to this edition and Mrs. Karin Bauer, who typed most of the copy. Finally, I would like to mention the cooperation I received from my publishing house. Bonn, July 1986 Karl-Rudolf Koch Preface to the First (German) Edition This book shall explain and substantiate the methods for estimating param eters, for testing hypotheses and for interval estimation. It therefore deals with the statistical inference for parameters. In order to address a large group of readers, representing different training and special fields, the book was planned such that no requirements are necessary for its understanding except for some basic acquaintance with the analysis. The knowledge which is needed from linear algebra and probability theory is presented in the first two chapters of the book. As it is the aim to discuss the inference for parameters with respect to the application, more emphasis was placed on conveying the understanding for the methods to be discussed, than on presenting general definitions and proofs. Linear models are applied for estimating the parameters and testing the hypotheses. This, however, does not mean a restriction of the generality, since under assumptions, which in general are readily fulfilled, the non-linear models can be transformed into linear models, as shall be shown. On the other hand, the linear models have the advantage that the methods of linear algebra can be applied. The definitions and theorems, which are needed, are therefore presented in Chapter 1. The theorems are proved with only a few exceptions. Chapter 1 also contains the generalized inverses, which are used for estimating parameters in models which are not of full rank, and presents the projections, which serve the geometrical interpretation of the estimation. Besides the estimation of parameters, which is applied to univariate and multivariate models, interval estimation, hypothesis testing and discriminant analysis are also treated. Hence, Chapter 2 deals with probability theory. The random variables are defined and the most important univariate and multivariate distributions are derived together with the test distributions for the univariate and multivariate models of estimating parameters. For each distribution a method is given to numerically compute its cumulative distribution function. Thus, printing tables of the distribution function could be dispensed with. Chapter 3 deals with a univariate and multivariate model to estimate parameters, which are defined as fixed quantities. Fixed parameters, together with random parameters, are estimated in the mixed model. The analysis of variance and the estimation of variance and covariance components are also presented in Chapter 3. Chapter 4 is devoted to the testing of hypotheses, to interval estimation and to testing for outliers, while Chapter 5 finally gives a short review of discriminant analysis. If theorems, which have been already presented, are needed for proofs, XIV Preface they will be referred to by means of the appropriate numbers. Hence, Chap ter 1 and also Chapter 2 do not need to be read before the remaining sections, since on the basis of the references to the theorems, the lacking knowledge can be found easily in the first two chapters. If a page number is given with a book cited as a reference, it only denotes the first page of interest; the following pages, which might also be of importance, are not pointed out. I want to thank all the co-workers in the Institute of Theoretical Geodesy, who contributed to the publication of this book. In particular, I thank Mr. Burkhard Schaffrin, Dipl.-Math., Dipl.-Ing., who made many suggestions. Fi nally, I would like to mention the cooperation I received from my publishing house during the preparation of the book. Bonn, September 1979 Karl-Rudolf Koch
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