Hjem

1 Introduction

2 Basic concepts of likelihood theory
2.1 Maximum likelihood estimation and information
2.2 Likelihood intervals and confidence intervals
2.3 Likelihood tests
2.4 Exercises

3 Exponential families
3.1 Motivation
3.2 Definition
3.3 Minimal representation and convex support
3.4 Laplace and cumulant transform
3.5 Estimation
3.6 Marginal and conditional distributions
3.7 Completeness of the minimal canonical statistic
3.8 Exercises

4 Suficiency
4.1 Introduction and definition
4.2 The suficiency principle
4.3 The factorization theorem
4.3.1 The case of a discrete sample space
4.3.2 The general case
4.4 Minimal suficient statistics
4.5 Exercises

5 Ancillarity and Basu's theorem
5.1 Definition and the conditionality principle
5.2 Suficiency and ancillarity
5.2.1 Two principles
5.2.2 Maximum likelihood estimation and information
5.3 Basu's theorem
5.4 Birnbaum's theorem and the likelihood principle
5.5 Exercises

6 Unbiased estimators with minimal variance and Cramér-Rao's inequal-
ity
6.1 Unbiased estimators with minimal variance
6.2 Variance inequalities
6.3 Suficiency, ancillarity and UMVU estimators
6.4 Exercises

7 Test theory
7.1 Introduction and definitions
7.2 Neyman-Pearson's lemma and monotone ratios
7.3 Composite null hypotheses { tests for a sub-parameter
7.4 Locally most powerful tests
7.5 Exercises

8 Inference on sub-parameters
8.1 Introduction
8.2 L-suficiency and L-ancillarity
8.3 S-suficiency and S-ancillarity
8.4 G-suficiency and G-ancillarity
8.5 Item analysis
8.6 Concluding remarks
8.7 Exercises

9 Bayesian statistics
9.1 Interpretations of probabilities
9.1.1 Laplace's classical interpretation
9.1.2 Objective probabilities
9.1.3 Probability as degree of belief
9.2 The Bayesian paradigm
9.3 Choice of the prior distribution
9.4 Bayesian asymptotics
9.5 Bayesian computation
9.6 Bayesian networks
9.7 Exercises

10 Decision theory
10.1 Basic setup and definitions
10.2 The likelihood ratio test
10.3 Bayes decision rules
10.4 Suficiency and decision rules
10.5 Exercises

11 Bibliographic notes and references

A Notation and useful results
A.1 Notation
A.2 The transformation theorem
A.3 Conditional expectation
A.4 Conditional densities
A.5 Formulae for densities and integrals
A.6 Inequalities
A.7 Uniqueness of the Laplace transform

Index