Sequential Analysis (October 24, 2013)

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序贯分析的作者wald写的教材。英文原版,书签带目录。 Sequential Analysis Paperback – October 24, 2013 by Abraham Wald (Author) In 1943, while in charge of Columbia University's Statistical Research Group, Abraham Wald devised Sequential Design, an innovative statistical inference system. Because the decision to terminate
DOVER PHOENIX EDITIONS opyright Copyright o 1947 by Abraham Wald Bibliographical note unaltered republication of the work originally published by Jonn Wiley &Sons, InC, in 1947 3d This Dover edition, first published in 1973 and reprinted in 2004, is an unabridged International standard book numnber eSBN-13.978-0-486-61579-0 Manufactured in the United States by Courier Corporation 615790012013 www.doverpublications.com PREFACE This book presents the theory of a recently developed method of statistical inference, that of sequential analysis. An effort has been made to keep the exposition on a level that will make most of the book with the exception of the Appendix, understandable to readers whose mathematical background does not go beyond college algebra and a first course in calculus. Some knowledge of probability and statistics is desirable for the understanding of the book, although not essential, for a brief review is given of the fundamental concepts, such as random variables, probability distributions, and statistical hypotheses To facilitate the reading of the book for those who have no advanced mathematical training, some concessions are made to generality and occasionally even to rigor Furthermore, mathematical derivations of somewhat intricate nature are put into the appendix, the reading of which may be omitted without impairing the understanding of the rest of the book This book contains an expanded exposition of the ideas and results i published in two technical papers on this subject, one of which appeared in 1944 and the other in 1945, as well as some further developments. Such developments, for example, are: the discussion of multivalued decisions and estimation in Part Ill; improvements in the limits for the average number of observations required by a sequential test; and limits for the effect of grouping in the binomial case. Some recent results of M. A. Girshick are included and, in the discussion of certain applications in Part ll, use is made of some simplifications contained in a publication of the statistical research Group of columbia University dealing with these applications Nearly all tables in the book were computed by the statistical research Group of columbia University while I was a consultant to the group. a few sections of my two forementioned publications have been incorporated in this book, mostly in the Appendix, without substantial changes I wish to express my indebtedness to Milton Friedman and W. Allen Wallis, who proposed the problem of sequential analysis to me in March, 1943. It was their clear formulation of the problem that gave me the incentive to start the investigations leading to the present developments. I also wish to express my thanks to the social Science research Council for their help which facilitated the publication of this book. I am indebted to Mr. mortimer Spiegelman of the Metropolitan Life Insurance Company for his careful reading of the manuscript and for making several valuable suggestions Thanks are due also to mrs. E. bowker who prepared the manuscript with particular care A W Columbia university March 1947 CONTENTS INTRODUCTION PARTL. GENERAL THEORY Chapter /. ELEMENTS OF THE CURRENT THEORY OF TESTING STATISTICAL HYPOTHESES 1 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. 1. 1 Notion of a random variable 1.1.2 Cumulative Distribution Function(c d.f. ) of a Random variable 1.1.3 Probability density function 1.1.4 Discrete random variables 1.1. 5 Expected value and Higher Moments of a Random variable 1. 2 NOTION OF A STATISTICAL HYPOTHESIS 1.2. 1 Unknown parameters of a distribution 1. 2.2 Simple and Composite Hypotheses 1. 3 OUTLINE OF THE CURRENT PROCEDURE FOR TESTING STATISTICAL HYPOTHESES 1.3.1 The Sample 1.3.2 The General Nature of a test procedure 1.3.3 Principles for Choosing a critical region 1.3.4 Number of Observations Necessary if a and B Have Preassigned Values 1.3.5 Testing a Hypothesis viewed as a Decision between Two Courses of action Chapter 2. SEQUENTIAL TEST OF A STATISTICAL HYPOTHESIS GENERAL DISCUSSION 2. 1 NOTION OF A SEQUENTIAL TEST 2.2 CONSEQUENCES OF THE CHOICE OF ANY PARTICULAR SEQUENTIAL TEST 2.2.1 The Operating characteristic function 2.2.2 The Average(Expected) Sample Number(ASn) Function of a Sequential test 2. 3 PRINCIPLES FOR THE SELECTION OF A SEQUENTIAL TEST 2.3. 1 Degree of Preference for Acceptance or Rejection of the Null Hypothesis Ho as a Function of the Parameter 6 2.3.2 Requirements Imposed on the oc function 2.3.3 The asn Function as a basis for the selection of a Sequential test 2. 4 THE CASE WHEN A SIMPLE HYPOTHESIS HO IS TESTED AGAINST A SINGLE ALTERNATIVE 2.4.1 Efficiency of a Sequential Test 2.4.2 Efficiency of the Current Test Procedure, Viewed as a particular Case of a Sequential Test Chapter 3. THE SEQUENTIAL PROBABILITY RATIO TEST FOR TESTING A SIMPLE HY POTHESIS Ho AGAINST A SINGLE ALTERNATIVE HI 3.1 DEFINITION OF THE SEQUENTIAL PROBABILITY RATIO TEST 3.2 FUNDAMENTAL RELATIONS AMONG THE QUANTITIES a, B, A, AND B 3. 3 DETERMINATION OF TIIE CONSTANTS A AND B IN PRACTICE 3.4 THE OC FUNCTION OF THE SEQUENTIAL PROBABILITY RATIO TEST 3.5 THE ASN FUNCTION OF A SEQUENTIAL PROBABILITY RATIO TEST 3.6 SAVING IN THE NUMBER OF OBSERVATIONS EFFECTED BY THE USE OF THE SEQUENTIAL PROBABILITY RATIO TEST INSTEAD OF THE CURRENT TEST PROCEDURE 3.7 LOWER LIMIT OF THE PROBABILITY THAT THE SEQUENTIAL TEST WILL TERMINATE WITII A NUMBER OF TRIALS LESS TIIAN OR EQUAL TO A GIVEN NUMBER 3.8 TRUNCATION OF THE SEQUENTIAL TEST PROCEDURE 3.9 INCREASE IN THE EXPECTED NUMBER OF OBSERVATIONS CAUSED BY REPLACING THE EXACT VALUES A (a, B)AND B(a, B)BY(1-B)/a AND B/( a), RESPECtiVely Chapter 4. OUTLINE OF A THEORY OF SEQUENTIAL TESTS OF SIMPLE AND COMPOSITE HYPOTHESES AGAINST A SET OF ALTERNATIVES 4.1 TESTS OF SIMPLE HYPOTHESES 4.1. 1 Introductory Remarks 4.1.3 Test of a simple hypothesis with No Restrictions on the es 4.1.2 Test of a Simple hypothesis against One-Sided alternativ Alternative Values of the unknown parameters 4.1.4 Application of the General Procedure to Testing the Mean of a Normal Distribution with Known variance 42 TESTS OF COMPOSITE HYPOTHESES 4.2. 1 Discussion of an Important Special Case 4.2.2 Outline of the test procedure in the general Case 4.2.3 Application of the General Procedure to Testing the Mean of a Normal Distribution with Unknown Variance(Sequential t-Test) 4.2. 4 A Particular Class of Problems Treated by girshick PART. APPLICATION OF THE GENERAL THEORY TO SPECIAL CASES Chapter 5. TESTING THE MEAN OF A BINOMIAL DISTRIBUTION (ACCEPTANCE INSPECTION OF A LOT WHERE EACH UNIT IS CLASSIFIED INTO ONE OF TWO CATEGORIES) 5.1 FORMULATION OF THE PROBLEM 5.2 TOLERATED RISKS OF MAKING WRONG DECISIONS 5.3 THE SEQUENTIAL PROBABILITY RATIO TEST CORRESPONDING TO THE QUANTITIES PO, Pl, a, AND B 5.3.1 Derivation of Algebraic Formulas for the Test Criterion 5.3.2 Tabular Procedure for Carrying Out the Test 5.3.3 Graphical Procedure for Carrying Out the Test 5.4 TIIE OPERATING CIIARACTERISTIC (OC) FUNCTION L(P) OF TIIE TEST 5.4.1 Determination of L(p) for Some Special Values of p 5.4.2 Determination of L(p) over the Whole Range ofp 5.4.3 Exact Formula for L(p) When the Reciprocal of the Slope of the Decision Lines Is an Integer 5.5 THE AVERAGE SAMPLE NUMBER (ASN) FUNCTION OF TIIE TEST 5.6 OBSERVATIONS TAKEN IN GROUPS 5.6. 1 General discussion 5.6.2 Upper and lower Limits for the Effect of Grouping on the oc and ASN Curves 5.7 TRUNCATION OF THE TEST PROCEDURE Chapter 6. TESTING THE DIFFERENCE BETWEEN THE MEANS OF TWO BINOMIAL DISTRIBUTIONS (DOUBLE DICHOTOMIES) 6. 1 FORMULATION OF THE PROBLEM 6.2 THE CLASSICAL METHOD 6.3 AN EXACT NON-SEQUENTIAL METHOD 6.4 SEQUENTIAL TEST OF TIE HYPOTIIESIS TIIATP12p2 6. 4. 1 Risks That We are Willing to Tolerate of Making Wrong D ecisions 6.4.2 The Sequential Probability Ratio Test Corresponding to the Quantities uo, u1, a, and B 6.4.3 The Operating Characteristic Curve of the Test 6. 4. 4 The average amount of Inspection Required by the test 6.4.5 Observations Taken in Groups Chapter 7. TESTING THAT THE MEAN OF A NORMAL DISTRIBUTION WITH KNOWN STANDARD DEVIATION FALLS SHORT OF A GIVEN VALUE 7.1 FORMULATION OF THE PROBLEM 7.2 TOLERATED RISKS OF MAKING WRONG DECISION 7.3 THE SEQUENTIAL PROBABILITY RATIO TEST CORRESPONDING TO THE QUANTITIES 80, 81, a AND B 7.4 THE OPERATING CHARACTERISTIC(OC) CURVE OF THE TEST 7.5 TIIE AVERAGE AMOUNT OF INSPECTION REQUIRED BY TIIE TEST Chapter 8. TESTING THAT THE STANDARD DEVIATION OF A NORMAL DISTRIBUTION DOES NOT EXCEED A GIVEN VALUE 8. 1 FORMULATION OF THE PROBLEM 8.2 TOLERATED RISKS FOR MAKING A WRONG DECISION 8.3 THE SEQUENTIAL PROBABILITY RATIO TEST CORRESPONDING TO THE QUANTITIES 0. 01. a. AND 8.4 THE OPERATING CHARACTERISTIC(OC) FUNCTION OF THE TEST 8.5 THE AVERAGE AMOUNT OF INSPECTION REQUIRED BY THE TEST 8.6 MODIFICATION OF THE TEST PROCEDURE WHEN THE POPULATION MEAN IS NOT KNOWN Chapter 9. TESTING THAT THE MEAN OF A NORMAL DISTRIBUTION WITH KNOWN VARIANCE IS EQUAL TO A SPECIFIED VALUE 9.1 FORMULATION OF THE PROBLEM

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