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INTRODUCTION TO PROBABILITY AND STATISTICS

STATISTICS: Textbooks and Monographs A Series Edited by D.

B. Owen, Founding Editor, 1972-1991

W. R. Schucany, Coordinating Editor Department o f Statistics Southern Methodist University Dallas, Texas

R. G. Cornell, Associate Editor for Biostatistics

W. J. Kennedy, Associate Editor for Statistical Computing

University o f Michigan

Iowa State University

A. M. Kshirsagar, Associate Editor for Multivariate Analysis and Experimental Design

E. G. Schilling, Associate Editor for Statistical Quality Control Rochester Institute o f Technology

University o f Michigan

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

The Generalized Jackknife Statistic, H. L. Gray and W. R. Schucany Multivariate Analysis, A n an t M . Kshirsagar Statistics and Society, W alter T. Federer Multivariate Analysis: A Selected and Abstracted Bibliography, 1 9 5 7 - 1 9 7 2 , Kocher/akota Subrahmaniam and Kathleen Subrahmaniam Design of Experiments: A Realistic Approach, Virgil L. Anderson and Robert A . McLean Statistical and M athem atical Aspects of Pollution Problems, John W. Pratt Introduction to Probability and Statistics (in tw o parts), Part I: Probability; Part II: Statistics, Narayan C. Giri Statistical Theory of the Analysis of Experimental Designs, J. O gaw a Statistical Techniques in Simulation (in tw o parts), Jack P. C. Kleijnen Data Quality Control and Editing, Joseph I. Naus Cost of Living Index Numbers: Practice, Precision, and Theory, Kali S. Banerjee Weighing Designs: For Chemistry, Medicine, Economics, Operations Research, Statistics, Kali S. Banerjee The Search for Oil: Some Statistical Methods and Techniques, edited by D. B. Owen Sample Size Choice: Charts for Experiments w ith Linear Models, Robert E. Odeh and M artin Fox Statistical Methods for Engineers and Scientists, Robert M . Bethea, Benjamin S. Duran, and Thomas L. BouIIion Statistical Quality Control M ethods, Irving W. Burr On the History of Statistics and Probability, edited by D. B. Owen Econometrics, Peter Schm idt

19. Sufficient Statistics: Selected Contributions, Vasant S. Huzurbazar fedited by A n a n t M . Kshirsagar) 2 0 . Handbook of Statistical Distributions, Jag dish K. Patel, C. H. Kapadia, and D. B. O w en 2 1 . Case Studies in Sample Design, A . C. Rosander 2 2 . Pocket Book of Statistical Tables, compiled by R. E. Odeh, D. B. O w en, Z. W. Birnbaum, and L. Fisher 2 3 . The Information in Contingency Tables, D. V. Gokhafe and Solomon Kullback 2 4 . Statistical Analysis of Reliability and Life-Testing Models: Theory and Methods, Lee J. Bain 2 5 . Elementary Statistical Quality Control, Irving W. Burr 2 6 . An Introduction to Probability and Statistics Using BASIC, Richard A . Groeneve/d 2 7 . Basic Applied Statistics, B. L. Raktoe and J. J . Hubert 2 8 . A Primer in Probability, Kathleen Subrahmaniam 2 9 . Random Processes: A First Look, R. Syski 3 0 . Regression Methods: A Tool for Data Analysis, Rudolf J. Freund and Paul D. M inton 3 1 . Randomization Tests, Eugene S. Edgington 3 2 . Tables for Normal Tolerance Limits, Sampling Plans and Screening, Robert E. Odeh and D. B. O wen 3 3 . Statistical Computing, William J. Kennedy, Jr., and Jam es E. Gentle 3 4 . Regression Analysis and Its Application: A Data-Oriented Approach, Richard F. Gunst and Robert L. Mason 3 5 . Scientific Strategies to Save Your Life, /. D. J. Bross 3 6 . Statistics in the Pharmaceutical Industry, edited by C. Ralph Buncher and JiaYeong Tsay 3 7 . Sampling from a Finite Population, J. Hajek 3 8 . Statistical Modeling Techniques, S. S. Shapiro 3 9 . Statistical Theory and Inference in Research, T. A . Bancroft and C.-P. Han 4 0 . Handbook of the Normal Distribution, Jagdish K. Patel and Campbell B. Read 4 1 . Recent Advances in Regression Methods, Hrishikesh D. Vinod and A m an Ul/ah 4 2 . Acceptance Sampling in Quality Control, Edward G. Schilling 4 3 . The Randomized Clinical Trial and Therapeutic Decisions, edited by Niels Tygstrup, John M Lachin, and Erik Juhl 4 4 . Regression Analysis of Survival Data in Cancer Chem otherapy, W alter H. Carter, Jr., Galen L. Wampler, and Donald M . Stablein 4 5 . A Course in Linear Models, A n an t M. Kshirsagar 4 6 . Clinical Trials: Issues and Approaches, edited by Stanley H. Shapiro and Thomas H. Louis 4 7 . Statistical Analysis of DNA Sequence Data, edited by B. S. Weir 4 8 . Nonlinear Regression Modeling: A Unified Practical Approach, David A. Ratk o w sk y 4 9 . Attribute Sampling Plans, Tables of Tests and Confidence Limits for Proportions, Robert E. Odeh and D. B. Owen 5 0 . Experimental Design, Statistical Models, and Genetic Statistics, edited by Klaus Hinke/mann 5 1 . Statistical Methods for Cancer Studies, edited by Richard G. Cornell 5 2 . Practical Statistical Sampling for Auditors, A rthu r J. Wilburn 5 3 . Statistical Methods for Cancer Studies, edited by Edw ard J. W egman and Jam es G. Smith

54. Self-Organizing M ethods in Modeling: GMDH Type Algorithms, edited by Stanley J. Farlow 5 5 . Applied Factorial and Fractional Designs, Robert A . McLean and Virgil L. Anderson 5 6 . Design of Experiments: Ranking and Selection, edited by Thomas J. Santner and A jit C. Tamhane 5 7 . Statistical Methods for Engineers and Scientists: Second Edition, Revised and Expanded, Robert M . Bethea, Benjamin S. Duran, and Thomas L. Bou/lion 5 8 . Ensemble Modeling: Inference from Small-Scale Properties to Large-Scale Sys tem s, A lan E. Gelfand and Crayton C. Walker 59 . Computer Modeling for Business and Industry, Bruce L. Bowerman and Richard T. O'Connell 6 0 . Bayesian Analysis of Linear Models, Lyle D. Broemeling 6 1 . Methodological Issues for Health Care Surveys, Brenda Cox and Steven Cohen 62 . Applied Regression Analysis and Experimental Design, Richard J. Brook and Gregory C. Arnold 6 3 . Statpal: A Statistical Package for Microcomputers —PC-DOS Version for the IBM PC and Compatibles, Bruce J. Chalm er and David G. Whitmore 6 4 . Statpal: A Statistical Package for Microcomputers —Apple Version for the II, II + , and lie, David G. Whitmore and Bruce J. Chalmer 6 5 . Nonparametric Statistical Inference: Second Edition, Revised and Expanded, Jean Dickinson Gibbons 6 6 . Design and Analysis of Experiments, Roger G. Petersen 67 . Statistical Methods for Pharmaceutical Research Planning, Sten W. Bergman and John C. Gittins 6 8 . Goodness-of-Fit Techniques, edited by Ralph B. D'Agostino and M ichael A . Stephens 6 9 . Statistical Methods in Discrimination Litigation, edited by D. H. Kaye and M ikel Aickin 70 . Truncated and Censored Samples from Normal Populations, Helm ut Schneider 71 . Robust Inference, M . L. Tiku, W. Y. Tan, and N. Balakrishnan 7 2 . Statistical Image Processing and Graphics, edited by Edward J. W egman and Douglas J. DePriest 7 3 . Assignment Methods in Combinatorial Data Analysis, Lawrence J. Hubert 7 4 . Econometrics and Structural Change, Lyle D. Broemeling and Hiroki Tsurumi 75 . Multivariate Interpretation of Clinical Laboratory Data, Adelin A lbert and Eugene K. Harris 76 . Statistical Tools for Simulation Practitioners, Jack P. C. Kleijnen 77 . Randomization Tests: Second Edition, Eugene S. Edgington 78 . A Folio of Distributions: A Collection of Theoretical Quantile-Quantile Plots, Edward B. Fowlkes 7 9 . Applied Categorical Data Analysis, Daniel H. Freeman, Jr. 8 0 . Seemingly Unrelated Regression Equations Models: Estimation and Inference, Virendra K. Srivastava and David E. A . Giles 8 1 . Response Surfaces: Designs and Analyses, Andre I. Khuri and John A . Cornell 8 2 . Nonlinear Parameter Estimation: An Integrated System in BASIC, John C. Nash and M ary Walker-Smith 8 3 . Cancer Modeling, edited by Jam es R. Thompson and Barry W. Brown 8 4 . M ixture Models: Inference and Applications to Clustering, Geoffrey J. McLachlan and Kaye E. Basford

8 5 . Randomized Response: Theory and Techniques, A rijit Chaudhuri and Rahul M ukerjee 8 6 . Biopharmaceutical Statistics for Drug Development, edited by Karl E. Peace 8 7 . Parts per Million Values for Estimating Quality Levels, Robert E. Odeh and D. B. O w en 88 . Lognormal Distributions: Theory and Applications, edited by Edwin L. Crow and Kunio Shimizu 89 . Properties of Estimators for the Gamma Distribution, K. O. Bow m an and L. R. Shenton 90 . Spline Smoothing and Nonparametric Regression, Randall L. Eubank 9 1 . Linear Least Squares Computations, R. W. Farebrother 92 . Exploring Statistics, Damaraju Raghavarao 9 3 . Applied Time Series Analysis for Business and Economic Forecasting, Sufi M . Nazem 9 4 . Bayesian Analysis of Time Series and Dynamic Models, edited by Jam es C. Spall 9 5 . The Inverse Gaussian Distribution: Theory, Methodology, and Applications, Raj S. Chhikara and J. Leroy Folks 9 6 . Parameter Estimation in Reliability and Life Span Models, A . Clifford Cohen and B etty Jones Whitten 9 7 . Pooled Cross-Sectional and Time Series Data Analysis, Terry E. Die/man 9 8 . Random Processes: A First Look, Second Edition, Revised and Expanded, R. Syski 9 9 . Generalized Poisson Distributions: Properties and Applications, P. C. Consul 10 0. Nonlinear Lp-Norm Estimation, Rene Gonin and A rthu r H. M oney 10 1. Model Discrimination for Nonlinear Regression Models, Dale S. Borowiak 1 0 2 . Applied Regression Analysis in Econometrics, H o w ard E. Doran 1 0 3 . Continued Fractions in Statistical Applications, K. O. Bow m an and L. R. Shenton 10 4. Statistical Methodology in the Pharmaceutical Sciences, Donald A . Berry 1 0 5 . Experimental Design in Biotechnology, Perry D. Haaland 10 6. Statistical Issues in Drug Research and Development, edited by Karl E. Peace 10 7. Handbook of Nonlinear Regression Models, David A . R atkow sky 10 8. Robust Regression: Analysis and Applications, edited by Kenneth D. Lawrence and J e ffre y L. A rthu r 10 9. Statistical Design and Analysis of Industrial Experiments, edited by Subir Ghosh 1 1 0 . (-/-Statistics: Theory and Practice, A . J. Lee 11 1. A Primer in Probability: Second Edition, Revised and Expanded, Kathleen Subrahmaniam 11 2. Data Quality Control: Theory and Pragmatics, edited by Gunar E. Liepins and V. R. R. Uppuluri 11 3. Engineering Quality by Design: Interpreting the Taguchi Approach, Thomas B. Barker 11 4. Survivorship Analysis for Clinical Studies, Eugene K. Harris and Adeiin Albert 1 1 5 . Statistical Analysis of Reliability and Life-Testing Models: Second Edition, L eeJ. Bain and M ax Enge/hardt 11 6. Stochastic Models of Carcinogenesis, Wai-Yuan Tan 11 7. Statistics and Society: Data Collection and Interpretation: Second Edition, Re vised and Expanded, Walter T. Federer 11 8. Handbook of Sequential Analysis, B. K. Ghosh and P. K. Sen 11 9 . Truncated and Censored Samples: Theory and Applications, A . Clifford Cohen 12 0. Survey Sampling Principles, E. K. Foreman

12 1. Applied Engineering Statistics, Robert M. Bethea and R. Russe/I Rhinehart 122. Sample Size Choice: Charts for Experiments w ith Linear Models: Second Edition, Robert E. Odeh and M artin Fox 1 2 3. Handbook of the Logistic Distribution, edited by N. Balakrishnan 1 2 4. Fundamentals of Biostatistical Inference, Chap T. Le 125. Correspondence Analysis Handbook, J.-P. Benz6cri 1 2 6. Quadratic Forms in Random Variables: Theory and Applications, A . M . M a th a i and Serge B. Provost 1 2 7. Confidence Intervals on Variance Components, Richard K. Burdick and Frankiin A . Gray bill 1 2 8. Biopharmaceutical Sequential Statistical Applications, edited by Karl E. Peace 12 9. Item Response Theory: Parameter Estimation Techniques, Frank B. Baker 130. Survey Sampling: Theory and M ethods, Arijit Chaudhuri and Horst Stenger 1 3 1. Nonparametric Statistical Inference: Third Edition, Revised and Expanded, Jean Dickinson Gibbons and Subhabrata Chakraborti 132. Bivariate Discrete Distribution, Subrahmaniam Kocherlakota and Kathleen Kocherlakota 133. Design and Analysis of Bioavailability and Bioequivalence Studies, Shein-Chung Chow and Jen-pei Liu 13 4. Multiple Comparisons, Selection, and Applications in Biometry, edited by Fred M . Hoppe 1 3 5. Cross-Over Experiments: Design, Analysis, and Application, David A . Ratkowsky, M arc A . Evans, and J. Richard Alldredge 136. Introduction to Probability and Statistics: Second Edition, Revised and Expanded, Narayan C. Giri

A dditional Volumes in Preparation

INTRODUCTION TO PROBABILITY AND STATISTICS Second Edition, Revised and Expanded

N arayan C. G iri Department o f Mathematics University o f M ontreal Montreal, Quebec, Canada

Marcel Dekker, Inc.

New York*Basel*Hong Kong

Library of Congress Cataloging-in-Publication Data Giri, Narayan C. Introduction to probability and statistics / Narayan C. Giri. ~ 2nd ed., rev. and expanded. p. cm. -- (Statistics, textbooks and monographs ; v. 136) Includes bibliographical references and index. ISBN 0-8247-9037-5 (alk. paper) 1. probabilities. 2. Mathematical statistics. I. Title. II. Series. QA273.G526 1993 519.5—dc20 93-12127 CIP

The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below. This book is printed on acid-free paper. Copyright © 1993 by Marcel Dekker, Inc.

All Rights Reserved.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Marcel Dekker, Inc. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

To my mother and the memories of my father and mother-in-law

Preface to the Second Edition

This is a revised and expanded edition of the original publication Intro duction to Probability and Statistics: Part 7, Probability, Part I I , Statistics. Nearly eighteen years have passed since the first edition of the book was published. During that time, undergraduate courses have changed greatly. Many first-year graduate topics are now included in undergraduate courses. This new edition attempts to bring the original book up to date by thorough revision, rewriting, addition of four new chapters, and addition of new problems and materials in each chapter. In preparing this volume I have tried to incorporate various comments by reviewers of the original book and by colleagues who have used it as a textbook and made numerous corrections to the original version. The comments of my own students and my long experience in teaching the course have also been utilized in the revision. This book is designed for a two-semester course in statistics, but can also be used for a one-semester course by suitable choice of chapters. Narayan C. Giri

Preface to the First Edition

PART I, PROBABILITY This book is intended to provide an introductory understanding of the theory of probability to students at the undergraduate level. Careful consideration has been given, in writing this text, to the needs of students with a special interest in statistics. It is hoped that the book will match the contents of courses on this subject for beginning and advanced undergraduates in most United States and Canadian universities. The basic definitions and ideas of the subject are covered, and an effort has been made to familiarize the student with fundamental results but not much other acquaintance with advanced mathematics will be found necessary. A thorough knowledge of calculus is assumed on the part of the reader. Some important results of matrix algebra have been included in the Appendix to supplement the readers’ knowledge in this area and to ensure a better understanding of the probabilistic treat ments of problems involving more than a single variable. Chapter 1 provides a little historical background of the theory of prob ability and discusses the importance of the various concepts in actual models in natural sciences from a logical point of view. General concepts, including both classical and axiomatic definitions and the usual basic theorems have been included in Chapter 2. Chapter 3 is actually a translation of some of the concepts introduced in the earlier chapters into the characteristics of

viii

Preface to the First Edition

actual models among natural phenomena. Further results and systematic study of these characteristics are included in the last two chapters. In keeping with the nature of an introductory text, many examples and motivations relevant to specific topics have been included. Exercises are given at the end of each section to make the mathematical treatment of the preceding theory more comprehensible. I feel that it would be appropriate to spread the material discussed in this book over two three-hour, one-semester courses, However, the first three chapters may be used in a three-hour, one-semester course in elementary probability theory; but Section 2.2 and parts of Chapter 3 should be avoided if such a course is intended particularly for students with a narrower mathe matical background than juniors in mathematics. If the reader finds this book useful, the credit is entirely due to my own teachers and colleagues, in particular Professors C. M. Stein, S. Karlin, and E. Parzen of Stanford University, Professor J. Kiefer of Cornell Uni versity, Professors H. K. Nandi and P. K. Bose of Calcutta University and Dr. A. K. Gayen of 1.1. T. Kharagpur, India, under whose influence I have come to appreciate the statistics of the present century. The preparation and revision of the manuscript of this book would not have been an easy task without the help of Professor A. R. Roy and Dr. S. K. Basu, who helped me by reading the entire manuscript with great care and diligence and offering valuable suggestions at various stages. The com ments of Professor Y. Lepage on Chapters 4 and 5 have been extremely helpful. I wish to express my gratitude and special thanks to all of them. I have presented the materials in parts in different courses in the Department of Mathematics, University of Montreal. The comments of the students have helped me to improve the presentation. I express my thanks to all of them. My wife Nilima, daughter Nabanita, and son Nandan have been very helpful and patient during the preparation of the book. I gratefully acknowl edge their assistance. The assistance of my parents and my elder brothers is also gratefully acknowledged. I would like to express my sincere thanks to the National Research Council of Canada and the Ministry of Education, Government of Quebec, for finan cial assistance during the preparation of the manuscript. Finally, I would like to express my gratitude to the secretaries of the Department of M athe matics, University of Montreal, for an excellent job in typing the manu script. Special thanks are due to the editors of Marcel Dekker, Inc., for put ting the original manuscript into its final form.

PART II, STATISTICS This book is intended to be a systematic presentation of the introductory theory of some topics of mathematical statistics. The discussions have been

Preface to the First Edition

ix

specially tailored to match, in general, the contents of regular and advanced undergraduate courses in most U. S. and Canadian universities. Thus a knowl edge of undergraduate mathematics, particularly of calculus, has been assumed for appreciating the mathematical treatments in the book; some important results of the matrix algebra have been included in Appendix A to supple ment the readers’ knowledge in this area. Chapter 1 provides a tour of the various types of statistical problems and then discusses the broad nature of statistical inference. Data reduction tech niques taking advantage of sufficiency are also included in this chapter. Prob abilistic tools for the study of order statistics have been developed in Chapter 2. Different methods of parametric fixed sample estimation and testing of hypotheses have been discussed in detail in Chapters 3 and 4, respectively. Chapter 5 is concerned with sequential procedures of hypotheses testing, while Chapter 6 deals with the nonparametric methods for this aspect of statistical inference. A systematic study of the general linear hypothesis and analysis of variance has been presented in Chapter 7. To maintain the general nature of introductory textbooks, we have tried to include many examples and motivations relevant to specific topics. Many exercises are also given at the end of each chapter to make the mathematical treatments of the preceding theory more comprehensible. We feel that it will be appropriate to spread the materials o f the book over two three-hour, one-semester basic courses in statistical inference for undergraduates. The first semester course may take care of Chapters 1, 2, 3, and part of Chapter 4, while the remaining portions of the book may be cov ered in the second semester. If the readers find this book useful, the credit is entirely due to my teachers and colleagues like Prof. C. M. Stein, Prof. S. Karlin, Prof. E. Parzen of Stanford University, Prof. J. Kiefer of Cornell University, and Prof. H. K. Nandi and Prof. P. K. Bose of Calcutta University, under whose influence I have come to appreciate the statistics of the present century. The preparation and the revision of the manuscript would not have been an easy task without the help of Dr. S. K. Basu who helped me by reading the entire manuscript with great care and diligence and offering valuable suggestions at various stages. I would like to express my gratitude and thanks to him. I have presented the materials in the book in parts in different courses in the Department of Mathematics, University of M ontreal. The comments of the students were very useful for improving the presentation. My wife Nilima, daughter Nabanita, and son Nandan have also been very helpful and patient during the preparation of this book. I gratefully acknowledge their assistance. The assistance of my mother and brothers is also gratefully ac knowledged. I would like to thank John Wiley and Sons, Inc., for their permission to reproduce Tables IB, 4B, and 5B from their publications and the editors of

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Preface to the First Edition

Biometrika for their permission to reproduce Table 6B. I would also like to express my sincere thanks to the National Research Council of Canada and the Ministry of Education, Government of Quebec, for financial assistance for the preparation of the manuscript. Finally, I would like to express my gratitude to the secretaries of the Department of Mathematics, University of Montreal, for an excellent job in typing the manuscript. Narayan C. Giri

Contents

Preface to the Second Edition Preface to the First Edition 1.

2.

vii

INTRODUCTION

1

1.1 1.2

Stochastic Model of Natural Phenomena Scientific Methodology and Statistics Bibliography

1 4 8

G EN ER A L CONCEPTS OF PROBABILITY

9

2.1 2.2 2.3

3.

v

Classical Definition of Probability Axiomatic Definition of Probability Bayes’ Theorem Bibliography

9 34 52 54

RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, AND CH ARACTERISTIC FUNCTIONS

55

3.1 3.2

55 58

Random Variables Discrete Univariate Random Variables

xi

xii

Contents 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

4.

Continuous Univariate Random Variables Multidimensional Random Variables Marginal Distributions of a Bivariate Random Variable Conditional Distributions of a Bivariate Random Variable Independence of Random Variables Functions of a Random Variable Functions of Several Random Variables Distribution of Product and Ratio of Two Random Variables Mathematical Expectation Moments Probability-Generating Functions Mathematical Expectation of a Function of a Bivariate Random Variable Conditional M athematical Expectation Two Im portant Inequalities Bibliography

STOCHASTIC CO N V ERG EN C E AND LIMIT THEOREM S 4.1 4.2

Two Types of Convergence Limit Theorems Bibliography

61 70 75 78 81 87 91 96 99 106 114 115 129 131 135

136 138 155 167

CONCEPTS OF STATISTICS

169

5.1 5.2 5.3 5.4 5.5 5.6

169 170 174 178 180 187 191

Inductive Inference The Nature of a Statistical Problem The Nature of Statistical Inference Transitions in the History of Statistical Methodology Data Reduction and Sufficiency The Exponential Family of Distributions Bibliography

UNIVARIATE DISTRIBUTIONS

192

6.1 6.2

193 204 210

Standard Univariate Distributions Sampling Distribution Bibliography

xiii

Contents 7.

8.

9.

MULTIVARIATE DISTRIBUTIONS

211

7.1 7.2 7.3 7.4 7.5 7.6 7.7

211 215 219 229 235 237 240 245

O R D E R STATISTICS AND RELATED DISTRIBUTIONS

246

8.1 8.2 8.3

246 247 251

Order Parameters Order Statistics Some Related Distributions

STATISTICAL INFERENCE: PARAM ETRIC POINT ESTIM ATION 9.1 9.2 9.3 9.4 9.5 9.6

10.

Properties of Multivariate Distributions Bivariate Normal Distribution Multivariate Normal Distribution Elliptically Symmetric Distributions Multinomial Distribution Distribution of Quadratic Forms Multivariate Normal Case: Distribution of Sample Mean and Sample Covariance Matrices Bibliography

Criteria for Judging Estimators Completeness and the Best Unbiased Estim ator Most Efficient Estimator and Consistent Estim ator Various Methods of Estimation Estimation of Parameters in Np(|x,£) Estimation of Parameters in Multinomial Population Bibliography

259 260 268 275 285 297 310 312

TESTING OF STATISTICAL HYPOTHESES

313

10.1 10.2 10.3 10.4

314 317 328

10.5 10.6 10.7 10.8 10.9

Basic Definitions Tests of Simple H0 Against Simple H x Tests of Simple H{) Against Composite H x Tests ofComposite Ha Against Composite H x (One-Parameter Case) Unbiased Tests Tests of Composite H n Against Composite H x (Multiparameter Case): Likelihood-Ratio Tests Test of Zero Correlation Confidence Intervals Tests and Confidence Interval for Mean in /Vp(|x,2) Bibliography

331 335 342 355 358 361 369

xiv 11.

12.

13.

Contents LARGE-SAM PLE M ETHODS

370

11.1 11.2 11.3 11.4 11.5

370 373 374 375 377 380

STATISTICAL DECISION TH EO R Y

381

12.1 12.2 12.3 12.4 12.5

381 383 384 390 396 400

15.

Basic Concepts Admissible Decision Rules Bayes’ Decision Rule Minimax Decision Rule Admissibility of Bayes’ Rules Bibliography

SEQ UEN TIAL ANALYSIS

401

13.1 13.2

402

13.3 13.4 13.5

14.

Introduction Edgeworth Approximation Variance Stabilizing Transformations Tests of Goodness of Fit Test of Independence in a Contingency Table Bibliography

Sequential Probability Ratio Tests Fundamental Relationship Between A , B and the Error Probabilities a , (3 Properties of the Stopping Rule N in the SPRT Operating Characteristic Function Average Sampling Number of the SPRT Bibliography

405 407 412 415 417

NO NPARA M ETRIC M ETHODS

418

14.1 14.2 14.3

418 420 428 431

One-Sample M ethods Two-Sample Methods Rank Test for the One-Way Classification Bibliography

G EN ER A L LIN EA R HYPOTHESIS AND ANALYSIS OF VARIANCE 15.1 15.2 15.3 15.4 15.5

Least Squares Estimates of (3 Maximum-Likelihood Estimates of (3 and