Zacks / Lakshmikantham / Tsokos Parametric Statistical Inference

Basic Theory and Modern Approaches
1. Auflage 2014
ISBN: 978-1-4831-5049-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 404 Seiten, Web PDF

Parametric Statistical Inference: Basic Theory and Modern Approaches presents the developments and modern trends in statistical inference to students who do not have advanced mathematical and statistical preparation. The topics discussed in the book are basic and common to many fields of statistical inference and thus serve as a jumping board for in-depth study. The book is organized into eight chapters. Chapter 1 provides an overview of how the theory of statistical inference is presented in subsequent chapters. Chapter 2 briefly discusses statistical distributions and their properties. Chapter 3 is devoted to the problem of sufficient statistics and the information in samples, and Chapter 4 presents some basic results from the theory of testing statistical hypothesis. In Chapter 5, the classical theory of estimation is developed. Chapter 6 discusses the efficiency of estimators and some large sample properties, while Chapter 7 studies the topics on confidence intervals. Finally, Chapter 8 is about decision theoretic and Bayesian approach in testing and estimation. Senior undergraduate and graduate students in statistics and mathematics, and those who have taken an introductory course in probability will highly benefit from this book.

Autoren/Hrsg.

Zacks, Shelemyahu

Lakshmikantham, V.

Tsokos, C. P.

Inhaltsverzeichnis

1;Front Cover;1
2;Parametric Statistical Inference: Basic Theory and Modern Approaches;4
3;Copyright Page;5
4;Table of Contents;10
5;Dedication;6
6;PREFACE;8
7;List of Illustrations;14
8;CHAPTER 1. GENERAL REVIEW;18
8.1;1.1 Introduction;18
8.2;1.2 Statistical Models, Distribution Functions and the Essence of Statistical Inference;19
8.3;1.3 The Information in Samples and Sufficient Statistics;21
8.4;1.4 Testing Statistical Hypotheses;22
8.5;1.5 Estimation Theory;23
8.6;1.6 The Efficiency of Estimators;25
8.7;1.7 Confidence and Tolerance Intervals;27
8.8;1.8 Decision Theoretic and Bayesian Approach in Testing and Estimation;28
9;CHAPTER 2 BASIC THEORY OF STATISTICAL DISTRIBUTIONS;32
9.1;2.1 Introductory Remarks;32
9.2;2.2 Elementary Properties of Distribution Functions;33
9.3;2.3 Some Families of Discrete Distributions;36
9.4;2.4 Some Families of Continuous Distributions;39
9.5;2.5 Expectations, Moments and Generating Functions;46
9.6;2.6 Joint Distributions, Conditional Distributions and Independence;50
9.7;2.7 Moments and Covariances of Linear Functions;60
9.8;2.8 Discrete Multivariate Distributions;63
9.9;2.9 Multinormal Distributions;66
9.10;2.10 Distributions of Symmetric Quadratic Forms of Normal Variables;71
9.11;2.11 Independence of Linear and Quadratic Forms of Normal Variables;74
9.12;2.12 The Order Statistics;76
9.13;2.13 t-Distributions
;78
9.14;2.14 F-Distributions
;79
9.15;2.15 The Distribution of the Sample Correlation;82
9.16;2.16 Limit Theorems;84
9.17;2.17 Problems;87
10;CHAPTER 3. SUFFICIENT STATISTICS AND THE INFORMATION IN SAMPLES;101
10.1;3.1 Introduction;101
10.2;3.2 Definitions and Characterization of Sufficient Statistics;102
10.3;3.3 Likelihood Functions and Minimal Sufficient Statistics;107
10.4;3.4 Sufficient Statistics and Exponential Type Families;112
10.5;3.5 Sufficiency and Completeness;116
10.6;3.6 Information Functions and Sufficiency;120
10.7;3.7 Problems;126
11;CHAPTER 4. TESTING STATISTICAL HYPOTHESES;130
11.1;4.1 The General Framework;130
11.2;4.2 The Neyman-Pearson Fundamental Lemma;135
11.3;4.3 Testing One-Sided Composite Hypotheses in MLR MOdels;140
11.4;4.4 Testing Two-Sided Hypotheses in One-Parameter Exponential Families;145
11.5;4.5 Testing Composite Hypotheses with Nuisance Parametens–Unbiased Tests
;147
11.6;4.6 Likelihood Ratio Tests;157
11.7;4.7 The Analysis of Contingency Tables;166
11.8;4.8 Sequential Testing of Hypotheses;171
11.9;4.9 Problems;182
12;CHAPTER 5. STATISTICAL ESTIMATION
;193
12.1;5. 1 General Discussion;193
12.2;5.2 Unbiased Estimators;194
12.3;5.3 Best Linear Unbiased and Least Squares Estfmators
;201
12.4;5.4 Stabilizing the Lse: Ridge Regression
;211
12.5;5.5 Maximum Likelihood Estimators
;215
12.6;5.6 Equivarlant Estimators;225
12.7;5.7 Moment-Equations Estimators;233
12.8;5.8 Pre-Test Estimators;237
12.9;5.9 ROBUST ESTIMATION OF THE LOCATION AND SCALE PARAMETERS OF SYMMETRIC DISTRIBUTIONS;239
12.10;5.10 Problems;242
13;CHAPTER 6. THE EFFICIENCY OF ESTIMATORS;253
13.1;6.1 General Introduction;253
13.2;6.2 The Cramér-Rao Lower Bound For The One-Parameter Case
;254
13.3;6.3 Extension of the Cramér-Rao Inequality to Multi-Parameter Cases;257
13.4;6.4 General Inequalities of the Cramér-Rao Type;259
13.5;6.5 The Efficiency of Estimators in Regular Cases
;261
13.6;6.6 Asymptotic Properties of Estimators;263
13.7;6.7 Second Order Asymptotic Efficiency;269
13.8;6.8 Maximum Probability Estimators;272
13.9;6.9 Problems;275
14;CHAPTER 7. CONFIDENCE AND TOLERANCE INTERVALS;279
14.1;7.1 General Introduction;279
14.2;7.2 The Construction of Confidence Intervals;281
14.3;7.3 Optimal Confidence Intervals;283
14.4;7.4 Large Sample Approximations;289
14.5;7.5 Tolerance Intervals;293
14.6;7.6 Distribution-Free Confidence and Tolerance Intervals;295
14.7;7.7 Simultaneous Confidence Intervals;298
14.8;7.8 Two-Stage and Sequential Sampling for Fixed Width Confidence Intervals
;302
14.9;7.9 Problems;306
15;CHAPTER 8. DECISION THEORETIC AND BAYESIAN APPROACH IN TESTING AND ESTIMATION;311
15.1;8.1 The Bayesian Framework;312
15.2;8.2 Bayesian Testing of Hypotheses;324
15.3;8.3 Bayesian Confidence Intervals;334
15.4;8.4 Bayes and Minimax Estimation
;338
15.5;8.5 Minimum Risk Equivariant, Formal Bayes and Structural Estimators;345
15.6;8.6 Empirical Bayes Estimators;356
15.7;8.7 The Admissibility of Estimators;359
15.8;8.8 Problems;373
16;REFERENCES;381
17;Author Index;397
18;Subject Index;401