Theory of preliminary test and Stein-type estimation with applications /

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Bibliographic Details
Author / Creator:	Saleh, A. K. Md. Ehsanes.
Imprint:	Hoboken, N.J. : Wiley ; Chichester : John Wiley [distributor], c2006.
Description:	xxiii, 622 p. : charts ; 24 cm.
Language:	English
Subject:	Parameter estimation. Regression analysis. Bayesian statistical decision theory. Bayesian statistical decision theory. Parameter estimation. Regression analysis.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/5933025

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ISBN:	0471563757 (hbk.)
Notes:	Includes bibliographical references (p. 601-612) and indexes.

Table of Contents:

List of Figures
List of Tables
Preface
1. Introduction
1.1. Objective of This Book
1.2. Statistical Decision Principle
1.3. Quadratic Loss Function
1.4. Some Statistical Models with Preliminaries
1.4.1. Mean and Simple Linear Models
1.4.2. One-Sample Multivariate Model
1.4.3. ANOVA Models
1.4.4. Parallelism Models
1.4.5. Multiple Regression Model and General Linear Hypothesis
1.4.6. Simple Multivariate Linear Model
1.4.7. Discrete Data Models
1.5. Organization of the Book
1.6. Conclusions
1.7. Problems
2. Preliminaries
2.1. Normal Distribution
2.2. Chi-square Distribution and Properties
2.3. Some Results from Multivariate Normal Theory
2.4. Beta Distribution and Applications
2.5. Discrete Distributions
2.5.1. Binomial Distribution
2.5.2. Multinomial Distribution
2.6. Matrix Results
2.7. Large Sample Theory
2.7.1. Four Types of Convergence
2.7.2. Law of Large Numbers
2.7.3. Central Limit Theorems
2.8. Nonparametric Theory: Preliminaries
2.8.1. Order-Statistics, Ranks, and Sign Statistics
2.8.2. Linear rank-statistics (LRS)
2.8.3. Rank Estimators of the Parameters of Various Models
2.9. Problems
3. Preliminary Test Estimation
3.1. Simple Linear Model, Estimators, and Tests
3.1.1. Simple Linear Model
3.1.2. Estimation of the Intercept and Slope Parameter
3.1.3. Test for the Slope Parameter
3.2. PTE of the Intercept Parameter
3.2.1. UE, RE and PTE of the Intercept Parameter
3.2.2. Bias and MSE Expressions
3.2.3. Comparison of bias and mse functions
3.2.4. Optimum Level of Significance of Preliminary Test
3.3. Two-Sample Problem and Pooling of Means
3.3.1. Model
3.3.2. Estimation and Test of the Difference between Two Means
3.3.3. Bias and mse Expression of the Three Estimators of a Mean
3.4. One-Sample Problem: Estimation of Mean
3.4.1. Model
3.4.2. Unrestricted, Restricted, and Preliminary Test Estimators
3.4.3. Bias, mse, and Analysis of Efficiency
3.5. An Alternative Approach
3.5.1. Introduction
3.5.2. One-Sample Problem
3.5.3. Comparison of PTE, [theta superscript PT subscript n] and SE [theta superscript S subscript n]
3.5.4. Simple Linear Model and Shrinkage Estimation
3.5.5. The Two-Sample Problem and Shrinkage Estimation
3.6. Estimation with Nonnormal Errors
3.6.1. Unrestricted, Restricted, Preliminary Test and Shrinkage Estimators, and the Test of Slope
3.6.2. Conditions for Asymptotic Normality of the Unrestricted Estimators of Intercept and Slope Parameters
3.6.3. Asymptotic Distributional Bias and Mean Square Error Expressions, and Efficiency Analysis
3.7. Two-Sample Problem and Estimation of Mean
3.8. One-Sample Problem and Estimation of the Mean
3.9. Stein Estimation of Variance: One-Sample Problem
3.10. Nonparametric Methods: R-Estimation
3.10.1. Model and Assumptions
3.10.2. Test of Hypothesis
3.10.3. Estimation of Intercept and Slope Parameters
3.10.4. Asymptotic Distribution of Various Estimators and Their ADB and ADMSE Expressions
3.11. Conclusions
3.12. Problems
4. Stein-Type Estimation
4.1. Statistical Model, Estimation, and Tests
4.2. Preliminary Test Estimation
4.3. Stein-Type Estimators
4.3.1. Introduction
4.3.2. James-Stein Estimator (JSE)
4.3.3. Positive-Rule Stein Estimator (PRSE)
4.3.4. Sclove-Morris-Radhakrishnan Modifications
4.4. Derivation of the Stein-Type Estimators
4.4.1. Risk Difference Representation Approach
4.4.2. Empirical Bayes Estimation (EBE) Approach
4.4.3. Quasi-empirical Bayes or Preliminary Test Estimation Approach
4.4.4. How Close is the JS Estimator to the Bayes Estimator?
4.5. Stein-Type Estimation When the Variance is Unknown
4.5.1. Introduction: Model, Estimators, and Tests
4.5.2. Preliminary Test and Stein-Type Estimators
4.5.3. Empirical Bayes Estimation When the Variance Is Unknown
4.5.4. Bias, MSE Matrices, and Risk Expressions
4.5.5. Risk Analysis of the Estimators
4.5.6. An Alternative Improved Estimator of [theta]
4.6. Stein-Type Estimation: Nonnormal Distributions
4.6.1. Model, Estimation, and Test
4.6.2. Preliminary Test (or Quasi-empirical Bayes) Approach to Stein-Type Estimation of the Mean Vector
4.6.3. Asymptotic Distributional Bias Vector, Quadratic Bias, MSE Matrix, and Risk Expressions of the Estimators
4.7. Improving James-Stein Estimator Toward Admissible Estimator
4.7.1. Introduction
4.7.2. Improving [theta superscript S subscript n] via PTE
4.7.3. Iterative PTE to Obtain an Admissible Estimator
4.7.4. Extension to the Case Where the Variance Is Unknown
4.8. Confidence Set Estimation Based on Stein-Type Estimators
4.8.1. Introduction
4.8.2. Properties of the Recentered Confidence Set Based on PRSE
4.8.3. Confidence Set Based on Preliminary Test Estimator
4.8.4. Asymptotic Theory of Recentered Confidence Sets and Domination of Positive-Rule Coverage Probability
4.9. Nonparametric Methods: R-Estimation
4.9.1. Model and Assumption
4.9.2. Test of Hypothesis
4.9.3. Estimation of the Location Parameter
4.9.4. ADB, ADQB, ADMSE, and ADQR of the Estimators of Location Parameters
4.9.5. Asymptotic Properties of Confidence Sets
4.10. Conclusions
4.11. Problems
5. Anova Model
5.1. Model, Estimation, and Tests
5.1.1. ANOVA model
5.1.2. Estimation of the Parameters of the One-Way ANOVA Model
5.1.3. Test of Equality of the Treatment Means
5.2. Preliminary Test Approach and Stein-Type Estimators
5.2.1. Preliminary Test Approach (or Quasi-empirical Bayes Approach)
5.2.2. Bayes and Empirical Bayes Estimators of Treatment Means
5.3. Bias, Quadratic Bias, MSE, and Risk Expressions
5.3.1. Bias Expressions
5.3.2. MSE Matrix and Risk Expressions
5.4. Risk Analysis and Risk Efficiency
5.4.1. Comparison of [theta subscript n] and [theta subscript n]
5.4.2. Comparison of [theta superscript PT subscript n] and [theta subscript n] ([theta subscript n])
5.4.3. Comparison of [theta superscript S subscript n], [theta superscript S+ subscript n], and [theta subscript n]
5.5. MSE Matrix Analysis and Efficiency
5.5.1. Comparison of [theta subscript n] and [theta subscript n]
5.5.2. Comparison of [theta superscript PT subscript n] Relative to [theta subscript n] and [theta subscript n]
5.5.3. Comparison of [theta subscript n] and [theta superscript S subscript n] ([theta superscript S subscript n] and [theta superscript S+ subscript n])
5.6. Improving the PTE
5.7. ANOVA Model: Nonnormal Errors
5.7.1. Estimation and Test of Hypothesis
5.7.2. Preliminary Test and Stein-Type Estimators
5.8. ADB, ADQB, ADMSE, and ADQR of the Estimators
5.8.1. Asymptotic Distribution of the Estimators under Fixed Alternatives
5.8.2. Asymptotic Distribution of the Estimators under Local Alternatives
5.8.3. ADB, ADQB, MSE-Matrices, and ADQR of [theta superscript PT subscript n] [theta superscript S subscript n] and [theta superscript S+ subscript n]
5.9. Confidence Set Estimation
5.9.1. Confidence Sets and Coverage Probabilities
5.9.2. Analysis of the Confidence Sets
5.10. Asymptotic Theory of Confidence Set Estimation
5.10.1. Asymptotic Representations of Normalized Estimators under Fixed Alternatives
5.10.2. Asymptotic Coverage Probability of the Confidence Sets under Local Alternatives
5.11. Nonparametric Methods: R-Estimation
5.11.1. Model, Assumptions, and Linear Rank Statistics (LRS)
5.11.2. Preliminary Test and Stein-Type Estimators
5.11.3. Asymptotic Distributional Properties of R-Estimators
5.11.4. ADB, ADQB, ADMSE, and ADQR
5.12. Conclusions
5.13. Problems
6. Parallelism Model
6.1. Model, Estimation, and Test of Hypothesis
6.1.1. Parallelism Model
6.1.2. Estimation of the Intercept and Slope Parameters
6.1.3. Test of Parallelism
6.2. Preliminary Test and Stein-Type Estimators
6.2.1. The Estimators of Intercepts and Slopes
6.2.2. Bayes and Empirical Bayes Estimators of Intercepts and Slopes
6.3. Bias, Quadratic Bias, MSE Matrices, and Risk Expressions
6.3.1. Unrestricted Estimators of [beta] and [theta]
6.3.2. Restricted Estimators of [beta] and [theta]
6.3.3. Preliminary Test Estimators of [beta] and [theta]
6.3.4. James-Stein-type Estimators of [beta] and [theta]
6.3.5. Positive-Rule Stein Estimators of [beta] and [theta]
6.4. Comparison of the Estimators of the Intercept Parameter
6.4.1. Bias Comparison of the Estimators of the Intercept Parameter
6.4.2. MSE-matrix Comparisons
6.4.3. Weighted Risk Comparisons of the Estimators
6.5. Estimation of the Regression Parameters: Nonnormal Errors
6.5.1. Unrestricted, Restricted, Preliminary Test, James-Stein and Positive-Rule Stein Estimators and Test of Hypothesis
6.5.2. Conditions for Asymptotic Properties of the Estimators and Their Distributions
6.5.3. Asymptotic Distributions of the Estimators
6.5.4. Expressions for ADB, ADQB, ADMSE, and ADQR of the Estimators
6.6. Asymptotic Distributional Risk Properties
6.6.1. Comparison of [theta subscript n] and [theta subscript n]
6.6.2. Comparison of [theta superscript PT subscript n] and [theta subscript n]([theta subscript n])
6.6.3. Comparison of [theta superscript S subscript n] and [theta subscript n]([theta subscript n])
6.6.4. Comparison of [theta superscript S subscript n] and [theta superscript PT subscript n]
6.6.5. Comparison of [theta superscript S+ subscript n] and [theta superscript S subscript n], [theta subscript n], [theta superscript PT subscript n]
6.7. Asymptotic Distributional MSE-matrix Properties
6.8. Confidence Set Estimation: Normal Case
6.8.1. Confidence Sets for the Slope Parameters
6.8.2. Analysis of Coverage Probabilities
6.8.3. Confidence Sets for the Intercept Parameters when [sigma superscript 2] is Known
6.9. Confidence Set Estimation: Nonnormal Case
6.10. Nonparametric Methods: R-Estimation
6.10.1. Model, Assumptions, and Linear Rank Statistics
6.10.2. R-Estimation and Test of Hypothesis
6.10.3. Estimation of the Intercepts [theta subscript alpha] and the Slope [beta subscript alpha]
6.10.4. Asymptotic Distribution of the R-Estimators of the Slope Vector
6.10.5. Asymptotic Distributional Properties of the R-Estimators of Intercepts
6.10.6. Confidence Sets for Intercept and Slope Parameters
6.11. Conclusions
6.12. Problems
7. Multiple Regression Model
7.1. Model, Estimation, and Tests
7.1.1. Estimation of Regression Parameters of the Model
7.1.2. Test of the Null Hypothesis, H[beta] = h
7.2. Preliminary Test and Stein-Type Estimation
7.2.1. Preliminary Test (or Quasi-empirical Bayes) Approach
7.2.2. Bayes and Empirical Bayes Estimators of the Regression Parameters
7.3. Bias, Quadratic Bias, MSE, and Quadratic Risks
7.3.1. Bias Expressions
7.3.2. MSE Matrices and Weighted Risks of the Estimators
7.4. Risk Analysis of the Estimators
7.5. MSE-Matrix Analysis of the Estimators
7.6. Improving the PTE
7.7. Multiple Regression Model: Nonnormal Errors
7.7.1. Introduction
7.7.2. Estimation of Regression Parameters and Test of the Hypothesis
7.7.3. Preliminary Test and Stein-Type Estimation
7.8. Asymptotic Distribution of the Estimators
7.8.1. Asymptotic Distribution of the Estimators under Fixed Alternatives
7.8.2. Asymptotic Distribution of the Estimators under Local Alternatives, and ADB, ADQB, ADMSE, and ADQR
7.8.3. ADQR Analysis
7.9. Confidence Set Estimation
7.9.1. Preliminaries
7.9.2. Confidence Sets and the Coverage Probabilities
7.9.3. Analysis of the Coverage Probabilities
7.10. Asymptotic Theory of Confidence Sets
7.10.1. Confidence Sets
7.10.2. Asymptotic Properties of Confidence Sets
7.11. Nonparametric Methods: R-Estimation
7.11.1. Linear Rank Statistics, R-Estimators and Confidence Sets
7.11.2. Asymptotic Distributional Properties of the R-estimators
7.11.3. Asymptotic Properties of the Recentered Confidence Sets Based on R-Estimators
7.12. Conclusions
7.13. Problems
8. Regression Model: Stochastic Subspace
8.1. The Model, Estimation, and Test of Hypothesis
8.1.1. The Model Formulation
8.1.2. Mixed Model Estimation
8.1.3. Test of Hypothesis
8.1.4. Preliminary Test and Stein-type Mixed Estimators
8.2. Bias, MSE, and Risks
8.2.1. Bias and Quadratic Bias Expressions
8.2.2. MSE Matrix and Risk Expressions
8.2.3. MSE Matrix Comparisons of the Estimators
8.2.4. Risk Comparisons of the Estimations
8.3. Estimation with Prior Information
8.3.1. Estimation of [beta subscript 1] and Test of H[subscript 0 beta subscript 0] = H[subscript 1 beta subscript 1]
8.3.2. The Mixed Estimators
8.3.3. Bias Expressions
8.3.4. MSE Matrix and Risk Expressions
8.4. Stochastic Subspace Hypothesis: Nonnormal Errors
8.4.1. Introduction
8.4.2. Estimation of the Parameters and Test of Hypothesis
8.4.3. Preliminary Test and Stein-type Estimators
8.5. Asymptotic Distribution of the Estimators
8.5.1. Asymptotic Distribution of the Estimators under Fixed Alternatives
8.5.2. Asymptotic Distribution of the Estimators under Local Alternatives
8.6. Confidence Set Estimation: Stochastic Hypothesis
8.7. R-Estimation: Stochastic Hypothesis
8.8. Conclusions
8.9. Problems
9. Ridge Regression
9.1. Ridge Regression Estimators
9.1.1. Ridge Regression with Normal Errors
9.1.2. Nonparametric Ridge Regression Estimators
9.2. Ridge Regression as Bayesian Regression Estimators
9.3. Bias Expressions
9.3.1. Bias Vector of [beta superscript PT subscript n] (k)
9.4. Covariance, MSE Matrix, and Risk Functions
9.5. Performance of Estimators
9.6. Estimation of the Ridge Parameter
9.7. Conclusions
9.8. Problems
10. Regression Models with Autocorrelated Errors
10.1. Simple Linear Model with Autocorrelated Errors
10.1.1. Estimation of the Intercept and Slope Parameters when [rho] is Known
10.1.2. Preliminary Test and S-Estimation of [beta] and [theta]
10.1.3. Estimation of the Intercept and Slope Parameters When Autocorrelation Is Unknown
10.2. Multiple Regression Model with Autocorrelation
10.2.1. Estimation of [beta] and Test of Hypothesis of H[beta] = h
10.2.2. Preliminary Test, James-Stein and Positive-Rule Stein-Type Estimators of [beta]
10.3. Bias, MSE Matrices, and the Risk of Estimators When [rho] Is Known
10.4. ADB, ADMSE, and ADQR of the Estimators ([rho] Unknown)
10.5. Estimation of Regression Parameters When [rho] Is Near Zero
10.5.1. Preliminary Test and Stein-Type Estimators (Chen and Saleh, 1993)
10.5.2. Design of Monte Carlo Experiment
10.5.3. Empirical Results and Conclusions
10.6. Estimation of Parameters of an Autoregressive Gaussian Process
10.6.1. Estimation and Test of Hypothesis
10.6.2. Asymptotic Theory of the Estimators and the Test-Statistics
10.6.3. ADB, ADMSE Matrices, and ADQR of the Estimators
10.7. R-Estimation of the Parameters of the AR[p]-Models
10.7.1. R-Estimation of the Parameters of the AR[p] Model
10.7.2. Tests of Hypothesis and Improved R-Estimators of [theta]
10.7.3. Asymptotic Bias, MSE Matrix, and Risks of the R-Estimators
10.8. R-Estimation of the Parameters with AR[1] Errors
10.9. Conclusions
10.10. Problems
11. Multivariate Models
11.1. Point and Set Estimation of the Mean Vector of an MND
11.1.1. Model, Estimation, and Test of Hypothesis
11.1.2. Bias, QB, MSE Matrix, and Weighted Risk Expressions of the Estimators
11.1.3. Risk and MSE Analysis of the Estimators
11.2. U-statistics Approach to Estimation
11.2.1. Asymptotic Properties of Point and Set Estimation under Fixed Alternatives
11.2.2. Asymptotic Properties of the Point and Set Estimation under Local Alternatives
11.3. Nonparametric Methods: R-estimation
11.3.1. Asymptotic Properties of the Point Estimators
11.3.2. Asymptotic Properties Confidence Sets
11.4. Simple Multivariate Linear Regression Model
11.4.1. Model, Estimation and Tests
11.4.2. Preliminary Test and Stein-Type Estimators
11.4.3. Bias, Quadratic Bias, MSE Matrices, and Risk Expressions of the Estimators
11.4.4. Two-Sample Problem and Estimation of the Means
11.4.5. Confidence Sets for the Slope and Intercept Parameters
11.5. R-estimation and Confidence Sets for Simple Multivariate Model
11.5.1. Introduction
11.5.2. Asymptotic Properties of the R-estimators
11.6. Conclusions
11.7. Problems
12. Discrete Data Models
12.1. Product of Bernoulli Models
12.1.1. Model, Estimation, and Test
12.1.2. Bayes and Empirical Bayes Estimation
12.1.3. Asymptotic Theory of the Estimators and the Test of Departure
12.1.4. ADB, ADQB, ADMSE, and ADQR of Estimators
12.1.5. Analysis of the Properties of Estimators
12.1.6. Baseball Data Analysis
12.1.7. Asymptotic Properties of Confidence Sets
12.2. Product Binomial Distributions
12.2.1. Introduction
12.2.2. Model, Estimation, and Test of Hypothesis
12.2.3. Asymptotic Theory of the Estimators and the Test-Statistics
12.2.4. ADB, ADQB, ADMSE, and ADQR of the Estimators
12.2.5. Estimation of Odds Ratio under Uncertain Zero Partial Association
12.2.6. Odds Ratios: Application to Meta-analysis of Clinical Trials
12.3. Product of Multinomial Models
12.3.1. The Product of Multinomial Models
12.3.2. Estimation of the Parameters
12.3.3. Test of Independence in an r x c Contingency Table
12.3.4. Preliminary Test and Stein-Type Estimators of the Cell Probabilities
12.3.5. Bayes and Empirical Bayes Method
12.3.6. Asymptotic Properties
12.3.7. Asymptotic Properties of the Estimators under Local Alternatives
12.3.8. Analysis of the Asymptotic Properties of the Estimators
12.4. Conclusions
12.5. Problems
References
Glossary
Authors Index
Subject Index

Theory of preliminary test and Stein-type estimation with applications /

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