Solutions to Selected Exercises

Mathematical Statistics
A Decision Theoretic Approach

Thomas S. Ferguson, UCLA

Published by Academic Press, New York, 1967.

• Chapter 1. Game Theory and Decision Theory
  • Section 1.4. Utility and Subjective Probability
  • Section 1.5. Randomization
  • Section 1.6. Optimal Decision Rules
  • Section 1.7. Geometric Interpretation for finite Parameter Space
  • Section 1.8. The Form of Bayes Rules for Estimation Problems

• Chapter 2. The Main Theorems of Decision Theory
  • Section 2.1. Admissibility and Completeness
  • Section 2.2. Decision Theory
  • Section 2.3. Admissibility of Bayes Rules
  • Section 2.4. Basic Assumptions
  • Section 2.5. Existence of Bayes Rules
  • Section 2.6. Existence of a Minimal Complete Class
  • Section 2.7. The separating Hyperplane Theorem
  • Section 2.8. Essential Completeness of the Class of Nonrandomized Decision Rules
  • Section 2.9. The Minimax Theorem
  • Section 2.10. The Complete Class Theorem
  • Section 2.11. Solving for Minimax Rules

• Chapter 3. Distributions and Sufficient Statistics
  • Section 3.1. Useful Univariate Distributions
  • Section 3.2. The Multivariate Normal Distribution
  • Section 3.3. Sufficient Statistics
  • Section 3.4. Essentially Complete Classes of Rules Based on Sufficient Statistics
  • Section 3.5. Exponential Families of Distributions
  • Section 3.6. Complete Sufficient Statistics
  • Section 3.7. Continuity of the Risk Function

• Chapter 4. Invariant Statistical Decision Problems
  • Section 4.1. Invariant Decision Problems
  • Section 4.2. Invariant Decision Rules
  • Section 4.3. Admissible and Minimax Invariant Rules
  • Section 4.5. Minimax Estimates of Location Parameters
  • Section 4.6. Minimax Estimates for Parameters of a Normal Distribution
  • Section 4.7. The Pitman Estimate
  • Section 4.8. Estimation of a Distribution Function

• Chapter 5. Testing Hypotheses
  • Section 5.1. The Neyman-Pearson Lemma
  • Section 5.2. Uniformly Most Powerful Tests
  • Section 5.3. Two-Sided Tests
  • Section 5.4. Uniformly Most Powerful Unbiased Tests
  • Section 5.5. Locally Best Tests
  • Section 5.6. Invariance in Hypothesis Testing
  • Section 5.7. The Two-Sample Problem
  • Section 5.8. Confidence Sets
  • Section 5.9. The General Linear Hypothesis
  • Section 5.10. Confidence Ellipsoids and Multiple Comparisons

• Chapter 6. Multiple Decision Problems
  • Section 6.1. Monotone Multiple Decision Problems
  • Section 6.2. Bayes Rules in Multiple Decision Problems
  • Section 6.3. Slippage Problems

• Chapter 7. Sequential Decision Problems
  • Section 7.2. Bayes and Minimax Sequential Decision Rules
  • Section 7.3. Convex Loss and Sufficiency
  • Section 7.4. Invariant Sequential Decision Problems
  • Section 7.7. The Fundamental Identity of Sequential Analysis