Mathematical Statistics and Stochastic Processes

Table of Contents

Preface

PART 1. MATHEMATICAL STATISTICS

Chapter 1. Introduction to Mathematical Statistics

1.1. Generalities

1.2. Examples of statistics problems

Chapter 2. Principles of Decision Theory

2.1. Generalities

2.2. The problem of choosing a decision function11

2.3. Principles of Bayesian statistics

2.4. Complete classes

2.5. Criticism of decision theory – the asymptotic point of view

2.6. Exercises

Chapter 3. Conditional Expectation

3.1. Definition

3.2. Properties and extension

3.3. Conditional probabilities and conditional distributions

3.4. Exercises

Chapter 4. Statistics and Sufficiency

4.1. Samples and empirical distributions

4.2. Sufficiency

4.3. Examples of sufficient statistics – an exponential model

4.4. Use of a sufficient statistic

4.5. Exercises

Chapter 5. Point Estimation

5.1. Generalities

5.2. Sufficiency and completeness

5.3. The maximum-likelihood method

5.4. Optimal unbiased estimators

5.5. Efficiency of an estimator

5.6. The linear regression model

5.7. Exercises

Chapter 6. Hypothesis Testing and Confidence Regions

6.1. Generalities

6.2. The Neyman–Pearson (NP) lemma

6.3. Multiple hypothesis tests (general methods)

6.4. Case where the ratio of the likelihoods is monotonic

6.5. Tests relating to the normal distribution

6.6. Application to estimation: confidence regions

6.7. Exercises

Chapter 7. Asymptotic Statistics

7.1. Generalities

7.2. Consistency of the maximum likelihood estimator

7.3. The limiting distribution of the maximum likelihood estimator ...