Sequential Estimation

The only comprehensive guide to the theory and practice of one of today's most important probabilistic techniquesThe past 15 years have witnessed many significant advances in sequential estimation, especially in the areas of three-stage and nonparametric methodology. Yet, until now, there were no references devoted exclusively to this rapidly growing statistical field.Sequential Estimation is the first, single-source guide to the theory and practice of both classical and modern sequential estimation techniques--including parametric and nonparametric methods. Researchers in sequential analysis will appreciate the unified, logically integrated treatment of the subject, as well as coverage of important contemporary procedures not covered in more general sequential analysis texts, such as: \* Shrinkage estimation \* Empirical and hierarchical Bayes procedures \* Multistage sampling and accelerated sampling procedures \* Time-sequential estimation \* Sequential estimation in finite population sampling \* Reliability estimation and capture-recapture methodologies leading to sequential tagging schemesAn indispensable resource for researchers in sequential analysis, Sequential Estimation is an ideal graduate-level text as well. Cover......Page 1 Title......Page 5 Contents......Page 9 Preface......Page 15 1.1 Introduction......Page 17 1.2 Some Sequential Sampling Schemes in Practice......Page 23 1.2.2 Hypergeometric Waiting-Time Distribution......Page 24 1.2.3 Capture-Mark-Recapture Procedures......Page 25 1.2.4 Time-Sequential Models......Page 26 1.2.5 Sequential Models in Reliability Problems......Page 27 1.3 Organization of This Book......Page 28 2.2 Martingales......Page 35 2.3 Stopping Times......Page 37 2.4 Martingale Inequalities and Identities......Page 40 2.5 Submartingale Convergence Theorems......Page 51 2.6 Martingale Central Limit Theorems......Page 56 2.7 Random Central Limit Theorems and Berry-Esseen Bounds......Page 60 2.8 Renewal Theorem First Passage and Residual Waiting Times......Page 66 2.9 Nonlinear Renewal Theory......Page 74 2.10 Exercises......Page 81 3.2 Decision-Theoretic Notions......Page 85 3.3 Bayesian Decision Rules......Page 89 3.4 Sufficiency and Efficiency......Page 91 3.5 Invariance and Transitivity......Page 97 3.6 Method of Maximum Likelihood......Page 98 3.7 Why Sequential?......Page 100 3.8 Exercises......Page 101 4.1 Introduction......Page 105 4.2 Sufficiency, Rao-Blackwell Theorem, and Transitivity......Page 106 4.3 Cramr-Rao and Related Inequalities......Page 112 4.4 Sequential Binomial Sampling Plans......Page 117 4.5 Exercises......Page 123 5.1 Introduction......Page 127 5.2 Bayesian Sequential Decision Rules......Page 128 5.3 Sequential Bayesian Estimation......Page 138 5.4 Asymptotically Pointwise Optimal (APO) Stopping Rules......Page 141 5.5 Hierarchical and Empirical Bayes Sequential Estimation......Page 154 5.6 Exercises......Page 166 6.2 Fixed-Width Confidence Intervals and Two-Stage Procedures......Page 169 6.2.1 Stein's Two-Stage Procedure......Page 170 6.2.2 Modified Two-Stage Procedure......Page 172 6.2.3 Further Generalizations......Page 173 6.3 Fixed-Width Confidence Intervals and Three-Stage Procedures......Page 175 6.3.1 The Global Theory......Page 176 6.3.2 Applications of the Three-Stage Procedure......Page 180 6.4 Fixed-Width Confidence Intervals and Accelerated Sequential Procedures......Page 184 6.4.1 The Global Theory......Page 185 6.5.1 Minimum Risk Normal Mean Problem......Page 189 6.5.2 Two-Stage Procedure......Page 190 6.5.4 Three-Stage Procedure......Page 191 6.5.5 Accelerated Sequential Procedure......Page 193 6.6.1 Point Estimation in Exponential Populations......Page 194 6.6.2 Estimation of Normal Variance......Page 198 6.6.3 Binomial and Negative Binomial Problems......Page 200 6.7.1 Fixed-Width Confidence Intervals......Page 201 6.7.2 Point Estimation......Page 204 6.8 Estimation in Multivariate Normal and Linear Models......Page 207 6.8.1 Estimation of Mean Vecior When S Is Arbitrary ......Page 208 6.8.3 Linear Regression Problems......Page 213 6.8.4 Shrinkage Estimators......Page 218 6.8.5 Estimation of Ordered Parameters......Page 219 6.9 Exercises......Page 220 7.1 Introduction......Page 227 7.2 Estimation of the Normal Mean......Page 228 7.3 Estimation of the Difference of Two Normal Means......Page 238 7.4 Point Estimation in Linear Models......Page 240 7.5 Estimation of the Multivariate Normal Mean......Page 243 7.6 Sequential Shrinkage Estimation......Page 248 7.7 Sequential Estimation of the Gamma Scale Parameter......Page 256 7.8 Exercises......Page 259 8.2 Fixed-Width Interval Estimation of the Normal Mean......Page 265 8.3 Sequential Interval Estimation of the Difference of Two Normal Means......Page 272 8.4 Fixed-Size Confidence Bounds for Linear Regression Parameters......Page 276 8.5 Confidence Region for the Mean Vector......Page 279 8.6 Exercises......Page 281 9.1 Introduction......Page 285 9.2 Estimable Parameters and MRE......Page 286 9.3 Differentiable Statistical Functionals and MRE......Page 303 9.4 Simple Semiparametric Models......Page 309 9.5 Multiparameter AMRE, I......Page 319 9.6 Multiparameter AMRE, II......Page 325 9.7 Exercises......Page 328 10.1 Introduction......Page 331 10.2 Type-A Confidence Intervals......Page 332 10.3 Type-B Confidence Intervals......Page 339 10.4 Nonparametric Confidence Sets......Page 344 10.5 Exercises......Page 348 11.2 Bias and Confidence Interval Evaluations......Page 351 11.2.1 Unknown Variance Case......Page 354 11.2.2 Another Practical Approach......Page 355 11.3 Sequential x2 and F Tests......Page 356 11.4 Exercises......Page 357 12.1 Introduction......Page 359 12.2 Time-Sequential Estimation for Poisson and Wiener Processes......Page 361 12.3 Time-Sequential Estimation for Exponential Life-Testing Models......Page 366 12.4 Some Generalizations......Page 375 12.5 Exercises......Page 380 13.1 Introduction......Page 383 13.2 Bundle Strength of Filaments......Page 384 13.3 System Reliability and Availability......Page 393 13.4 Sequential Estimation of Functional Parameters......Page 399 13.5 Exercises......Page 406 14.1 Introduction......Page 409 14.2 The CMRR and Two-Sample Estimators of N......Page 410 14.3 The CMRR and Multisample Estimators of N......Page 413 14.4 Estimation of N Under Inverse Sampling Schemes......Page 421 14.5 Sequential Tagging Schemes......Page 423 14.6 Bounded Percentage Width Confidence Interval for N......Page 428 14.7 Asymptotically Optimal Sequential Point Estimation of N......Page 434 14.8 Exercises......Page 437 15.1 Introduction......Page 441 15.2 General Asymptotics......Page 442 15.3 Sequential Perspectives......Page 447 15.4 Exercises......Page 459 References......Page 461 Author Index......Page 485 Subject Index......Page 489 The only comprehensive guide to the theory and practice of one oftoday's most important probabilistic techniques The past 15 years have witnessed many significant advances insequential estimation, especially in the areas of three-stage andnonparametric methodology. Yet, until now, there were no referencesdevoted exclusively to this rapidly growing statisticalfield. Sequential Estimation is the first, single-source guide to thetheory and practice of both classical and modern sequentialestimation techniques--including parametric and nonparametricmethods. Researchers in sequential analysis will appreciate theunified, logically integrated treatment of the subject, as well ascoverage of important contemporary procedures not covered in moregeneral sequential analysis texts, such as: • Shrinkage estimation • Empirical and hierarchical Bayes procedures • Multistage sampling and accelerated sampling procedures • Time-sequential estimation • Sequential estimation in finite population sampling • Reliability estimation and capture-recapture methodologiesleading to sequential tagging schemes An indispensable resource for researchers in sequential analysis,Sequential Estimation is an ideal graduate-level text as well. This Book Devoted To Sequential Estimation Presents The Advances Of The Past Fifteen Years Including Those In The Areas Of Three--stage Accelerated Sequential Sampling Procedures. It Integrates The Diversities In Sequential Estimation In A Logical, Treating Both Classical And Modern Techniques And Including Parametric And Nonparametric Methods. Introduction And Coverage -- Probabilistic Results In Sequential Analysis -- Some Basic Concepts For Fixed-sample Estimation -- General Aspects Of Sequential Estimation -- Sequential Bayesian Estimation -- Multistage Estimation -- Parametric Sequential Point Estimation -- Parametric Sequential Confidence Estimation -- Nonparametric Sequential Point Estimation -- Nonparametric Sequential Confidence Estimation -- Estimation Following Sequential Tests -- Time-sequential Estimation Problems -- Sequential Estimation In Reliability Models -- Sequential Estimation Of The Size Of A Finite Population -- Stochastic Approximation Malay Ghosh, Nitis Mukhopadhyay, Pranab K. Sen. Includes Bibliographical References (p.445-468) And Index. The only comprehensive guide to the theory and practice of one of today's most important probabilistic techniquesThe past 15 years have witnessed many significant advances in sequential estimation, especially in the areas of three-stage and nonparametric methodology. Yet, until now, there were no references devoted exclusively to this rapidly growing statistical field. Sequential Estimation is the first, single-source guide to the theory and practice of both classical and modern sequential estimation techniques--including parametric and nonparametric methods. Researchers in sequential analysis This book devoted to sequential estimation presents advances of the past fifteen years including those in the areas of three-stage accelerated sequential sampling procedures. It integrates the diversities in sequential estimation in a logical, treating both classical and modern techniques and including parametric and nonparametric methods.Discusses shrinkage estimation, reliability sampling and capture-recapture methodology leading to sequential tagging schemes.-- Contains detailed appendices and indices.