Sequential Estimation

by Ghosh, Malay; Mukhopadhyay, Nitis; Sen, Pranab Kumar

• ISBN13: 9780471812715
• ISBN10: 0471812714
• Edition: 1st
• Format: Hardcover
• Copyright: 1997-03-14
• Publisher: Wiley-Interscience

Summary

The only comprehensive guide to the theory and practice of one of today's most important probabilistic techniques The 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 schemes An indispensable resource for researchers in sequential analysis, Sequential Estimation is an ideal graduate-level text as well.

Author Biography

MALAY GHOSH, PhD, is a professor in the University of Florida Department of Statistics. NITIS MUKHOPADHYAY, PhD, is a professor in the University of Connecticut Department of Statistics. PRANAB K. SEN, PhD, is a professor in the Department of Biostatistics and Statistics of the University of North Carolina.

Table of Contents

Preface	xiii
1. Introduction and Coverage	1	(18)
1.1 Introduction	1	(6)
1.2 Some Sequential Sampling Schemes in Practice	7	(5)
1.2.1 Binomial Waiting-Time Distribution	8	(1)
1.2.2 Hypergeometric Waiting-Time Distribution	8	(1)
1.2.3 Capture-Mark-Recapture Procedures	9	(1)
1.2.4 Time-Sequential Models	10	(1)
1.2.5 Sequential Models in Reliability Problems	11	(1)
1.2.6 Recursive Estimation and Sequential Schemes	12	(1)
1.3 Organization of This Book	12	(7)
2. Probabilistic Results in Sequential Analysis	19	(50)
2.1 Introduction	19	(1)
2.2 Martingales	19	(2)
2.3 Stopping Times	21	(3)
2.4 Martingale Inequalities and Identities	24	(11)
2.5 Submartingale Convergence Theorems	35	(5)
2.6 Martingale Central Limit Theorems	40	(4)
2.7 Random Central Limit Theorems and Berry-Esseen Bounds	44	(6)
2.8 Renewal Theorem-First Passage and Residual Waiting Times	50	(8)
2.9 Nonlinear Renewal Theory	58	(7)
2.10 Exercises	65	(4)
3. Some Basic Concepts for Fixed-Sample Estimation	69	(20)
3.1 Introdution	69	(1)
3.2 Decision-Theoretic Notions	69	(4)
3.3 Bayesian Decision Rules	73	(2)
3.4 Sufficiency and Efficiency	75	(6)
3.5 Invariance and Transitivity	81	(1)
3.6 Method of Maximum Likelihood	82	(2)
3.7 Why Sequential?	84	(1)
3.8 Exercises	85	(4)
4. General Aspectss of Sequential Estimation	89	(22)
4.1 Introduction	89	(1)
4.2 Sufficiency, Rao-Blackwell Theorem, and Transitivity	90	(6)
4.3 Cramer-Rao and Related Inequalities	96	(5)
4.4 Sequential Binomial Sampling Plans	101	(6)
4.5 Exercises	107	(4)
5. Sequential Bayesian Estimation	111	(42)
5.1 Introduction	111	(1)
5.2 Bayesian Sequential Decision Rules	112	(10)
5.3 Sequential Bayesian Estimation	122	(3)
5.4 Asymptotically Pointwise Optimal (APO) Stopping Rules	125	(13)
5.5 Hierarchical and Empirical Bayes Sequential Estimation	138	(12)
5.6 Exercises	150	(3)
6. Multistage Estimation	153	(58)
6.1 Introduction	153	(1)
6.2 Fixed-Width Conmfidence Intervals and Two-Stage Procedures	153	(6)
6.2.1 Stein's Two-Stage Procedures	154	(2)
6.2.2 Modified Two-Stage Procedure	156	(1)
6.2.3 Further Generalizations	157	(2)
6.3 Fixed-Width Confidence Intervals and Three-Stage Procedure	159	(9)
6.3.1 The Global Theory	160
6.3.2 Applications of The Three-Stage Procedure	164	(4)
6.4 Fixed-Width Confidence Intervals and Accelerated Sequential Procedure	168	(5)
6.4.1 The Global Theory	169	(4)
6.5 Point Estimation Problems	173	(5)
6.5.1 Minimum Risk Normal Mean Problem	173	(1)
6.5.2 Two-Stage Procedure	174	(1)
6.5.3 Modified Two-Stage Procedure	175	(1)
6.5.4 Three-Stage Procedure	175	(2)
6.5.5 Accelerated Sequantial Procedure	177	(1)
6.6 Other Related Estimation Problems	178	(7)
6.6.1 Point Estimation in Exponential Populations	178	(4)
6.6.2 Estimation of Normal Variance	182	(2)
6.6.3 Binomial and Negative Binomial Problems	184	(1)
6.7 Comparison of Populations	185	(6)
6.7.1 Fixed-Width Confidence Intervals	185	(3)
6.7.2 Point Estimation	188	(3)
6.8 Estimation in Multivariate Normal and Linear Models	191	(13)
6.8.1 Estimation of Mean Vector When Is Arbitrary	192	(5)
6.8.2 Comparison of Populations	197	(1)
6.8.3 Linear Regression Problems	197	(5)
6.8.4 Shrinkage Estimators	202	(1)
6.8.5 Estimation of Ordered Parameters	203	(1)
6.9 Exercises	204	(7)
7. Parametric Sequential Point Estimation	211	(38)
7.1 Introduction	211	(1)
7.2 Estimation of the Normal Mean	212	(10)
7.3 Estimation of the Difference of Two Normal Means	222	(2)
7.4 Point Estimation of Linear Models	224	(3)
7.5 Estimation of the Multivariate Normal Mean	227	(5)
7.6 Sequantial Shrinkage Estimation	232	(8)
7.7 Sequential Estimation oof the Gamma Scale Parameter	240	(3)
7.8 Exercises	243	(6)
8. Parametric Sequential Confidence Estimation	249	(20)
8.1 Introduction	249	(1)
8.2 Fixed-Width Interval Estimation of the Normal Mean	249	(7)
8.3 Sequential Interval Estimation of the Differece of Two Normal Means	256	(4)
8.4 Fixed-Size Confidence Bounds for Linear Regression Parameters	260	(3)
8.5 Confidence Region for the Mean Vector	263	(2)
8.6 Exercises	265	(4)
9. Nonparametric Sequential Point Estimation	269	(46)
9.1 Introduction	269	(1)
9.2 Estimable Parameters and MRE	270	(17)
9.3 Differentiable Statistical Functionals and MRE	287	(6)
9.4 Simple Semiparametric Models	293	(10)
9.5 Multiparameter AMRE, I	303	(6)
9.6 Multiparameter AMRE, II	309	(3)
9.7 Exercises	312	(3)
10. Nonparametric Sequential Confidence Estimation	315	(20)
10.1 Introduction	315	(1)
10.2 Type-A Confidence Intervals	316	(7)
10.3 Type-B Confidence Intervals	323	(5)
10.4 Nonparametric Confidence Sets	328	(4)
10.5 Exercises	323	(3)
11. Estimation Following Sequential Tests	335	(8)
11.1 Introduction	335	(1)
11.2 Bias and Confidence Interval Evaluations	335	(5)
11.2.1 Unknown Variance Case	338	(1)
11.2.2 Another Practical Approach	339	(1)
11.3 Sequential x 2 and F Tests	340	(1)
11.4 Exercises	341	(2)
12. Time-Sequential Estimation Problems	343	(24)
12.1 Introduction	343	(2)
12.2 Time-Sequential Estimation for Poisson and Wiener Processes	345	(5)
12.3 Time-Sequential Estimation for Exponential Life-Testing Models	350	(9)
12.4 Some Generalizations	359	(5)
12.5 Exercises	364	(3)
13. Sequential Estimation in Reliability Models	367	(26)
13.1 Introduction	367	(1)
13.2 Bundle Strength of Filaments	368	(9)
13.3 System Reliability and Availability	377	(6)
13.4 Sequential Estimation of Functional Parameters	383	(7)
13.5 Exercises	390	(3)
14. Sequential Estimation of the Size of a Finite Population	393	(32)
14.1 Introduction	393	(1)
14.2 The CMRR and Two-Sample Estimators of N	394	(3)
14.3 The CMRR and Multisample Estimators of N	397	(8)
14.4 Estimation of N Under Inverse Sampling Schemes	405	(2)
14.5 Sequential Tagging Schemes	407	(5)
14.6 Bounded Percentage Width Confidence Invetval for N	412	(6)
14.7 Asymptotically Optimal Sequential Point Estimation of N	418	(3)
14.8 Exercises	421	(4)
15. Stochastic Approximation	425	(20)
15.1 Introduction	425	(1)
15.2 General Asymptotics	426	(5)
15.3 Sequential Perspectives	431	(12)
15.4 Exercises	443	(2)
References	445	(24)
Author Index	469	(4)
Subject Index	473