Mathematical Statistics (eBook)

A Decision Theoretic Approach

Thomas S. Ferguson (Autor)

Z. W. Birnbaum, E. Lukacs (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
408 Seiten
Elsevier Science (Verlag)
978-1-4832-2123-6 (ISBN)

Infos

Mathematical Statistics: A Decision Theoretic Approach presents an investigation of the extent to which problems of mathematical statistics may be treated by decision theory approach. This book deals with statistical theory that could be justified from a decision-theoretic viewpoint. Organized into seven chapters, this book begins with an overview of the elements of decision theory that are similar to those of the theory of games. This text then examines the main theorems of decision theory that involve two more notions, namely the admissibility of a decision rule and the completeness of a class of decision rules. Other chapters consider the development of theorems in decision theory that are valid in general situations. This book discusses as well the invariance principle that involves groups of transformations over the three spaces around which decision theory is built. The final chapter deals with sequential decision problems. This book is a valuable resource for first-year graduate students in mathematics.

Inhalt

Front Cover 1
Mathematical Statistics: A Decision Theoretic Approach 4
Copyright Page 5
Table of Contents 10
Preface 6
CHAPTER 1. 
14 
1.1 Basic Elements 14
1.2 A Comparison of Game Theory and Decision Theory 18
1.3 Decision Function Risk Function
1.4 Utility and Subjective Probability 24
1.5 Randomization 35
1.6 Optimal Decision Rules 41
1.7 Geometric Interpretation for Finite 47
1.8 The Form of Bayes Rules for Estimation Problems 56
CHAPTER 2. 
67 
2.1 Admissibility and Completeness 67
2.2 Decision Theory 69
2.3 Admissibility of Bayes Rules 72
2.4 Basic Assumptions 76
2.5 Existence of Bayes Decision Rules 80
2.6 Existence of a Minimal Complete Class 82
2.7 The Separating Hyperplane Theorem 83
2.8 Essential Completeness of the Class of Nonrandomized Decision Rules 89
2.9 The Minimax Theorem 94
2.10 The Complete Class Theorem 99
2.11 Solving for Minimax Rules 103
CHAPTER 3. 
111 
3.1 Useful Univariate Distributions 111
3.2 The Multivariate Normal Distribution 118
3.3 Sufficient Statistics 125
3.4 Essentially Complete Classes of Rules Based on Sufficient 
132 
3.5 Exponential Families of Distributions 138
3.6 Complete Sufficient Statistics 145
3.7 Continuity of the Risk Function 150
CHAPTER 4. 
156 
4.1 Invariant Decision Problems 156
4.2 Invariant Decision Rules 161
4.3 Admissible and Minimax Invariant Rules 167
4.4 Location and Scale Parameters 177
4.5 Minimax Estimates of Location Parameters 179
4.6 Minimax Estimates for the Parameters of a Normal Distribution 189
4.7 The Pitman Estimate 199
4.8 Estimation of a Distribution Function 204
CHAPTER 5. 
211 
5.1 The Neyman-Pearson Lemma 211
5.2 Uniformly Most Powerful Tests 219
5.3 Two-Sided Tests 228
5.4 Uniformly Most Powerful Unbiased Tests 237
5.5 Locally Best Tests 248
5.6 Invariance in Hypothesis Testing 255
5.7 The Two-Sample Problem 263
5.8 Confidence Sets 270
5.9 The General Linear Hypothesis 277
5.10 Confidence Ellipsoids and Multiple Comparisons 287
CHAPTER 6. Multiple Decision Problems 297
6.1 Monotone Multiple Decision Problems 297
6.2 Bayes Rules in Multiple Decision Problems 304
6.3 Slippage Problems 312
CHAPTER 7. 
322 
7.1 Sequential Decision Rules 322
7.2 Bayes and Minimax Sequential Decision Rules 326
7.3 Convex Loss and Sufficiency 342
7.4 Invariant Sequential Decision Problems 353
7.5 Sequential Tests of a Simple Hypothesis Against a Simple Alternative 363
7.6 The Sequential Probability Ratio Test 374
7.7 The Fundamental Identity of Sequential Analysis 383
References 401
Index 406

Produkt-Details

• Erscheint lt. Verlag: 10.7.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• ISBN-10: 1-4832-2123-7 / 1483221237
• ISBN-13: 978-1-4832-2123-6 / 9781483221236