Sequential Statistics This page intentionally left blank Zakhula Gouindarajulu University of Kentucky, USA Seq uential Statistics \: World Scientific NEW JERSEY LONDON SINGAPORE BElJlNG SHANGHAI HONG KONG TAIPEI CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK once: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. SEQUENTIAL STATISTICS Copyright 0 2004 by World ScientificPublishing Co. Re. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-905-9 Printed in Singapore. “Education without wisdom and wisdom without humility are husks without grain.” Sri Sathya Sai Baba This page intentionally left blank Dedicated To the Memory of My parents This page intentionally left blank Preface Sequential statistics is concerned with treatment of data when the number of ob- servations is not fixed in advance. Since Wald (1947) wrote his celebrated book, the subject has grown considerably, especially in the areas of sequential estima- tion and biostatistics. The author’s book THE SEQUENTIAL STATISTICAL ANALYSIS OF HYPOTHESIS TESTING, POINT AND INTERVAL ESTIMA- TION, AND DECISION THEORY (available from American Sciences Press, Inc., 20 Cross Road, Syracuse, New York 13224-2104, U.S.A.), ISBN 0-935950-17-6, is a comprehensive 680 page reference to the field of sequential analysis; every- one reading the present new work will likely want to see that there is at least a copy of that comprehensive book in their institution’s library, and many serious researchers will want a copy in their personal libraries. Of that previous book, reviewers said “There are plenty of examples and problems” and “The presenta- tion is clear and to the point.” In contrast, the present new book is designed for a semester’s course and is thus less than half the length of the previous book. Other books by Ghosh (1970), Siegmund (1985)) Wetherill and Glazebrook (1986) and Ghosh, Mukhopadhyay and Sen (1997) are either too theoretical or limited in scope. It is so easy for an instructor to get side-tracked and bogged down with details and running out of time to cover interesting topics. In this new version, I have tried to select only those topics that can be covered in a semester’s course. Still, the instructor may not be able to cover all the topics in the book in one semester. Thus he has some flexibility in the choice of topics. Straightforward and elementary proofs are provided and for more details the reader is referred to the earlier book of the author. Thus, the mathematical and statistical level of the book is maintained at an elementary level. This book is geared to seniors and first year graduate students who have had a semester’s course in each of advanced calculus, probability and statistical inference. A semester’s course can be based on chapter 1, chapter 2 (excluding section 2.7) chapter 3 (excluding sections 3.7 and 3.8) chapter 4 ( excluding section 4.9) and chapter 5 (excluding sections 5.5 and 5.6). The instructor might devote three 50-minute lectures to chapter 1, ten lectures to chapter 2, nine lectures to each of chapter 3 and 4, and five lectures in chapter 5, with the remaining lectures devoted to sections of his/her and student’s interests. 1x X PREFACE The chapter on applications to biostatistics is new and the supplement con- taining computer programs to certain selected sequential procedures is also pro- vided. Useful illustrations and numerical tables are provided wherever possible. Problems identified by the section to which they pertain are given at the ends of all chapters. An extensive list of references that are cited in the book is given at the end. This list of references is by no means complete. April 2004 Z. Govindarajulu Professor of Statistics University of Kentucky Lexington, KY Acknowledgments I have been inspired by the celebrated book on this topic by the late Abraham Wald and I am grateful to Dover Publications for their kind permission for my use of certain parts of Wald’s book as a source. I am equally grateful to the American Sciences Press for permission to use several sections of my earlier book as a source for the present book. I am grateful to Dr. Hokwon Cho of the University of Nevada, Las Vegas for putting the entire manuscript on LaTex and cheerfully making all the necessary changes in the subsequent revisions. I am thankful to the Department of Statistics for its support and other help. I thank Professors Rasul Khan of Cleveland State University and Derek S. Coad of the University of Sussex (England) for their useful comments on an earlier draft of the manuscript. I also wish to express my thanks to Ms. Yubing Zhai and Ms. Tan Rok Ting, editors of World Scientific Publishing Co. for her encouragement, cooperation and help. My thanks go to the American Statistical Association for its kind permission to reproduce table 3.9.3 and tables 3.9.1 & 3.9.2 from their publications, namely, the Journal of American Statistical Association Vol. 65 and the American Sta- tistician Vol. 25, respectively. To Blackwell Publishing Ltd. for its kind permission to reproduce tables 2.7.1 and 2.7.2 from the Journal of Royal Statistical Society Series B Vol. 20 and tables 3.2.18~3.2.2 and table 3.3.lfrom the Australian Journal of Statistics , Vols 31 and 36 respectively. To the University of Chicago Press for their kind permission to reproduce data set from Olson & Miller, Morphological Integration, p. 317 and to have brief excerpt from Kemperman, J. H. B. (1961)) the Passage Problem for a stationary Markov Chain. To Springer-Verlag GMbH & Co. to use Theorem 8.25 and Corollary 8.33 of Siegmund, D. 0.)(1985) Sequential Analysis as a source for Theorem 3.4.4 of the present book. To Professor Erich Lehmann for his kind permission to use his book Testing Statistical Hypotheses (1959) as a source for the proof of Theorem 2.9.1 and the xi xii ACKNOWLEDGMENTS statement of Theorem 2.9.2. To Oxford University Press for its permission of the Biometrika Trustees to reproduce Table 1 of Lindley and Barnett (1965) Biometriku, Vol. 52, p. 523. To Professors Donald Darling and Mrs. Carol Robbins for their kind permis- sion for reproducing Table 3.8.1 from Proceedings of the Nut. Acud. Sciences Vol. 60. To Professor Thomas Ferguson for his kind permission to use his distribution as Problem 2.1.6. To CRC Press for its kind permission to use Sections 1.2 and 1.3 of B. Wether- ill (1975) as a source for sections 1.1 and 1.2 of this book. To the Institute of Mathematical Statistics for its kind permission to repro- duce Tables 2.6.1, 2.10.1, 2.10.2 and 3.8.1 from the Annuls ofMuthemutica1 Sta- tistics and Annals of Statistics. To Francis Taylor Group for their kind permission to reproduce tables 5.4.1 and 5.4.2 from Stu&stics Vol. 33. To John Wiley & Sons, for their kind permission to use Whitehead (1983) sections 3.7 and 3.8 as a source for section 5.6 of this book. Contents Preface ix Acknowledgments xi 1 Preliminaries 1 1.1 Introduction to Sequential Procedures ................ 1 1.2 Sampling Inspection Plans ....................... 3 1.2.1 Sample Size Distribution .................... 3 1.3 Stein’s Two-stage Procedure ...................... 6 1.3.1 The Procedure ......................... 7 2 The Sequential Probability Ratio Test 11 2.1 The Sequential Probability Ratio Test (SPRT) ........... 11 2.2 SPRT: It’s Finite Termination and Bounds ............. 13 2.3 The Operating Characteristic Function ................ 19 2.4 The Average Sample Number ..................... 21 2.5 Wald’s F’undamental Identity ..................... 29 2.5.1 Applications of the F‘undamental Identity .......... 30 2.6 Bounds for the Average Sample Number ............... 33 2.7 Improvements to OC and ASN Functions .............. 36 2.7.1 The OC F‘unction ........................ 36 2.7.2 The Average Sample Number ................. 38 2.8 TruncatedSPRT ............................ 40 2.9 Optimal Properties of the SPRT ................... 45 2.10 The Restricted SPRT ......................... 47 2.11 Large-Sample Properties of the SPRT ................ 51 2.12 Problems ................................ 54 3 Tests for Composite Hypotheses 59 3.1 Method of Weight Functions ...................... 59 3.1.1 Applications of the Method of Weight Functions ...... 60 3.2 Sequential t and t2 Tests ....................... 61 xiii XiV CONTENTS 3.2.1 Uniform Asymptotic Expansion and Inversion for an Integral 63 3.2.2 Barnard’s Versions of Sequential t- and t2-tests ....... 65 3.2.3 Simulation Studies ....................... 65 3.2.4 Asymptotic Normality of the Statistic T ........... 66 3.2.5 Finite Sure Termination of Sequential t- and t2-tests .... 69 3.2.6 Sequential t2-test (or t-test for Two-sided Alternatives) . 71 3.2.7 The Sequential Test T ..................... 73 3.2.8 An Alternative Sequential Test T’ .............. 74 3.3 Sequential F-test ............................ 75 3.3.1 Inversion Formula ......................