Robust Statistical Methods
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690 William J. J. Rey
Robust Statistical Methods
Springer-Verlag Berlin Heidelberg New York 1978
Author William J. J. Rey MBLE - Research Laboratory 2, Avenue van Becelaere B-1170 Brussels
Library of Congress Cataloging in Publication Data
1940J Rey, William J. Robust statistical methods. 690) (Lecture notes in mathematics Bibliography: p. Includes indexes. 1. Robust statistics. 2. Nonparametric statistics. 3. Estimation theory. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 690. QA3.L28 no. 690 CQA276J 510'.8s C519.5J 78-24262
AMS Subject Classifications (1970): Primary: 62G35 Secondary: 62G25, 62J05 ISBN 3-540-09091-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09091-6 Springer-Verlag New York Heidelberg Berlin 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 fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
FOREWORD During the last nine years, several problems in the statistical processing of biomedical data have been encountered by the author. These problems had in common the fact that most of the usual assumptions were without any solid basis. Poor quality samples drawn from unknown distributions, usually non-normal and frequently non-stationary, were the ordinary lot; nevertheless, sophisticated parameters had to be reliably estimated and the scatter of the estimators was needed to permit comparison of the results. been partly solved by application of robust methods.
This has
The methods presented in this text are oriented toward the design of robust estimators. The primary concern is preventing any significant offset of the estimates due to the selection of an erroneous model or to spurious data in the sample. and variance estimation.
The second concern is bias reduction The special emphasis reserved to type M
estimators is justified by their analytical form which permits to assess their properties, even for small sample sizes (n=10 or 50), without resorting to involved arguments. The theoretical tools are mainly the jackknife and the influence function. Applied derivations are in the fields of location estimation and regression analysis. aspects.
Due attention is devoted to computational
TABLE OF CONTENTS 1, INTRODUCTION 1.1. History and main contributions 1.2. Why robust estimations ? 1 • 3. Summary 2, ON SAMPLING DISTRIBUTIONS
2. 1. Scope of section 2.2. Metrics for probability distributions
8 8 8
2.3. Definition of robustness, breakdownpoint
11
2.4. Estimators seen as functional of distributions
12
2.5. The influence function of Hampel
15
3. TH
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