Ferguson / Birnbaum / Lukacs Mathematical Statistics

A Decision Theoretic Approach
1. Auflage 2014
ISBN: 978-1-4832-2123-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 408 Seiten, Web PDF

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.

Autoren/Hrsg.

Ferguson, Thomas S.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Statistics: A Decision Theoretic Approach;4
3;Copyright Page;5
4;Table of Contents;10
5;Preface;6
6;CHAPTER 1.
Game Theory and Decision Theory;14
6.1;1.1 Basic Elements;14
6.2;1.2 A Comparison of Game Theory and Decision Theory;18
6.3;1.3 Decision Function; Risk Function;19
6.4;1.4 Utility and Subjective Probability;24
6.5;1.5 Randomization;35
6.6;1.6 Optimal Decision Rules;41
6.7;1.7 Geometric Interpretation for Finite;47
6.8;1.8 The Form of Bayes Rules for Estimation Problems;56
7;CHAPTER 2.
The Main Theorems of Decision Theory;67
7.1;2.1 Admissibility and Completeness;67
7.2;2.2 Decision Theory;69
7.3;2.3 Admissibility of Bayes Rules;72
7.4;2.4 Basic Assumptions;76
7.5;2.5 Existence of Bayes Decision Rules;80
7.6;2.6 Existence of a Minimal Complete Class;82
7.7;2.7 The Separating Hyperplane Theorem;83
7.8;2.8 Essential Completeness of the Class of Nonrandomized Decision Rules;89
7.9;2.9 The Minimax Theorem;94
7.10;2.10 The Complete Class Theorem;99
7.11;2.11 Solving for Minimax Rules;103
8;CHAPTER 3.
Distributions and Sufficient Statistics;111
8.1;3.1 Useful Univariate Distributions;111
8.2;3.2 The Multivariate Normal Distribution;118
8.3;3.3 Sufficient Statistics;125
8.4;3.4 Essentially Complete Classes of Rules Based on Sufficient
Statistics;132
8.5;3.5 Exponential Families of Distributions;138
8.6;3.6 Complete Sufficient Statistics;145
8.7;3.7 Continuity of the Risk Function;150
9;CHAPTER 4.
Invariant Statistical Decision Problems;156
9.1;4.1 Invariant Decision Problems;156
9.2;4.2 Invariant Decision Rules;161
9.3;4.3 Admissible and Minimax Invariant Rules;167
9.4;4.4 Location and Scale Parameters;177
9.5;4.5 Minimax Estimates of Location Parameters;179
9.6;4.6 Minimax Estimates for the Parameters of a Normal Distribution;189
9.7;4.7 The Pitman Estimate;199
9.8;4.8 Estimation of a Distribution Function;204
10;CHAPTER 5.
Testing Hypotheses;211
10.1;5.1 The Neyman-Pearson Lemma;211
10.2;5.2 Uniformly Most Powerful Tests;219
10.3;5.3 Two-Sided Tests;228
10.4;5.4 Uniformly Most Powerful Unbiased Tests;237
10.5;5.5 Locally Best Tests;248
10.6;5.6 Invariance in Hypothesis Testing;255
10.7;5.7 The Two-Sample Problem;263
10.8;5.8 Confidence Sets;270
10.9;5.9 The General Linear Hypothesis;277
10.10;5.10 Confidence Ellipsoids and Multiple Comparisons;287
11;CHAPTER 6. Multiple Decision Problems;297
11.1;6.1 Monotone Multiple Decision Problems;297
11.2;6.2 Bayes Rules in Multiple Decision Problems;304
11.3;6.3 Slippage Problems;312
12;CHAPTER 7.
Sequential Decision Problems;322
12.1;7.1 Sequential Decision Rules;322
12.2;7.2 Bayes and Minimax Sequential Decision Rules;326
12.3;7.3 Convex Loss and Sufficiency;342
12.4;7.4 Invariant Sequential Decision Problems;353
12.5;7.5 Sequential Tests of a Simple Hypothesis Against a Simple Alternative;363
12.6;7.6 The Sequential Probability Ratio Test;374
12.7;7.7 The Fundamental Identity of Sequential Analysis;383
13;References;401
14;Index;406