Sequential statistics

Annotation This book contains topics that can be covered in a single-semester course. Only elementary proofs are provided, and thus the mathematics and statistics are maintained at a basic level. Only a course in each of three areas - advanced calculus, probability and statistical inference - is assumed of the student. The book has a chapter on applications to biostatistics and a supplement presenting computer programs for selected sequential procedures. Identified problems are provided at the end of each chapter Cover......Page 1 Sequential Statistic - Title......Page 4 Preface......Page 10 Acknowledgements......Page 12 Contents......Page 14 1.1 Introduction to Sequential Procedures......Page 18 1.2.1 Sample Size Distribution......Page 20 1.3 Stein's Two-stage Procedure......Page 23 1.3.1 The Procedure......Page 24 2.1 The Sequential Probability Ratio Test (SPRT)......Page 28 2.2 SPRT: It's Finite Termination and Bounds......Page 30 2.3 The Operating Characteristic Function......Page 36 2.4 The Average Sample Number......Page 38 2.5 Wald's Fundamental Identity......Page 46 2.5.1 Applications of the Fundamental Identity......Page 47 2.6 Bounds for the Average Sample Number......Page 50 2.7.1 The OC Function......Page 53 2.7.2 The Average Sample Number......Page 55 2.8 Truncated SPRT......Page 57 2.9 Optimal Properties of the SPRT......Page 62 2.10 The Restricted SPRT......Page 64 2.11 Large-Sample Properties of the SPRT......Page 68 2.12 Problems......Page 71 3.1 Method of Weight Functions......Page 76 b) Sequential Chi-square Test......Page 77 3.2 Sequential t and t2 Tests......Page 78 3.2.1 Uniform Asymptotic Expansion and Inversion for an Integral......Page 80 3.2.3 Simulation Studies......Page 82 3.2.4 Asymptotic Normality of the Statistic T......Page 83 3.2.5 Finite Sure Termination of Sequential t- and t2-tests......Page 86 3.2.6 Sequential t2-test (or t-test for Two-sided Alternatives)......Page 88 3.2.7 The Sequential Test T......Page 90 3.2.8 An Alternative Sequential Test T'......Page 91 3.3 Sequential F-test......Page 92 3.3.1 Inversion Formula......Page 94 3.4 Likelihood Ratio Test Procedures......Page 96 3.4.1 Generalized Likelihood Ratio Tests for Koopman-Darmois Families......Page 103 3.5.1 Armitage-Sobel-Wald Test......Page 107 OC Function......Page 110 3.5.3 Bounds for ASN......Page 111 3.5.4 Testing Two-sided Alternatives for Normal Mean......Page 113 3.6.1 Efficiency of the SPRT Relative to the Fixed-Sample Size Procedure at the Hypotheses......Page 116 3.6.2 Relative Efficiency at......Page 119 3.6.3 Limiting Relative Efficiency of the SPRT......Page 122 3.7.1 Bayes Sequential Binomial SPRT......Page 123 The Methods of Calculation......Page 127 3.7.2 Dynamic Programming Method for the Binomial Case......Page 128 3.7.3 The Dynamic Programming Equation......Page 129 3.7.4 Bayes Sequential Procedures for the Normal Mean......Page 131 3.8 Small Error Probability and Power One Test......Page 134 Tests with small error probability......Page 136 Tests with Power One......Page 137 3.9.1 Kolmogorov-Smirnov Tests with Power One......Page 140 3.9.2 Sequential Sign Test......Page 141 3.9.3 Rank Order SPRT's Based on Lehmann Alternatives: Two-Sample Case......Page 144 3.9.4 One-Sample Rank Order SPRT's for Symmetry......Page 147 One-sided Alternatives......Page 152 Two-sided Alternatives......Page 153 3.10 Appendix: A Useful Lemma......Page 155 3.11 Problems......Page 156 4.1 Basic Concepts......Page 160 4.2 Sufficiency and Completeness......Page 161 4.3 Cramer-Rao Lower Bound......Page 169 4.4.1 Stein's Procedure for Estimating the Mean of a Normal Distribution with Unknown Variance......Page 175 Normal Populations......Page 179 4.4.3 Procedures for Estimating the Common Mean......Page 181 A Simpler Estimator......Page 183 4.4.4 Double-Sampling Estimation Procedures......Page 184 Estimation with given Variance: Single unknown Parameter......Page 185 Estimation in the presence of a Nuisance Parameter......Page 187 Confidence Intervals......Page 188 4.4.5 Fixed Length Confidence Intervals Based on SPRT......Page 190 kth step......Page 192 Applications......Page 194 Special Case......Page 196 4.5 Large-Sample Theory for Estimators......Page 199 Special Cases......Page 201 4.6 Determination of Fixed-width Intervals......Page 208 4.7.1 Interval Estimation for the Mean......Page 213 4.7.2 Risk-efficient Estimation of the Mean......Page 218 4.8.1 Fixed-Size Confidence Bounds......Page 220 When is Unknown......Page 221 Asymptotic Properties of the Class C......Page 222 4.9 Confidence Intervals for P(X