统计理论

Preface

Chapter 1: Probability Models

1.1 Background

1.1.1 General Concepts

1.1.2 Classical Statistics

1.1.3 Bayesian Statistics

1.2 Exchangeability

1.2.1 Distributional Symmetry

1.2.2 Frequency and Exchangeability

1.3 Parametric Models

1.3.1 Prior, Posterior, and Predictive Distributions

1.3.2 Improper Prior Distributions

1.3.3 Choosing Probability Distributions

1.4 DeFinetti's Representation Theorem

1.4.1 Understanding the Theorems

1.4.2 The Mathematical Statements

1.5 Proofs of DeFinetti's Theorem and Related Results

1.5.1 Strong Law of Large Numbers

1.5.2 The Bernoulli Case

1.5.3 The General Finite Case

1.5.4 The General Infinite Case

1.5.5 Formal Introduction to Parametric Models

1.6 Infinite-Dimensional Parameters

1.6.1 Dirichlet Processes

1.6.2 Tailfree Processes

Chapter 2: Sufflcient Statistics

2.1.1 Notational Overview

2.1.3 Minimal and Complete Sufficiency

2.1.4 Ancillarity

2.2 Exponential Families of Distributions

2.2.1 Basic Properties

2.2.2 Smoothness Properties

2.2.3 A Characterization Theorem

2.3.1 Fisher Information

2.3.2 Kullback-Leibler Information

2.3.3 Conditional Information

2.4 Extremal Families

2.4.1 The Main Results

2.4.2 Examples

Chapter 3: Decision Theory

3.1 Decision Problems

3.1.2 Elements of Bayesian Decision Theory

3.1.3 Elements of Classical Decision Theory

3.1.4 Summary

3.2 Classical Decision Theory

3.2.1 The Role of Sufficient Statistics

3.2.2 Admissibility

3.2.3 James-Stein Estimators

3.2.5 Complete Classes

3.3 Axiomatic Derivation of Decision Theory

3.3.1 Definitions and Axioms

3.3.2 Examples

3.3.3 The Main Theorems

3.3.4 Relation to Decision Theory

3.3.5 Proofs of the Main Theorems

3.3.6 State-Dependent Utility

Chapter 4: Hypothesis Testing

4.1 Introduction

4.1.1 A Special Kind of Decision Problem

4.1.2 Pure Significance Tests

4.2 Bayesian Solutions

4.2.1 Testing in General

4.2.2 Bayes Factors

4.3 Most Powerful Tests

4.3.1 Simple Hypotheses and Alternatives

4.3.2 Simple Hypotheses, Composite Alternatives

4.3.3 0ne-Sided Tests

4.3.4 Two-Sided Hypotheses

4.4 Unbiased Tests

4.4.1 General Results

Chapter 5: E8timation

Chapter 6: Equivariance

Chapter 7: Large Sample Theory

Chapter 8: Hierarchical Models

chapter 9: sequential analysis

appendix a: measure and integration theory

appendix b: probability theory

appendix c: mathematical theorems not proven here

appendix d: summary of distributions

references

notation and abbreviation index

name index

subject index

《数学与金融经典教材：统计理论（影印版）》是一部经典的讲述统计理论的研究生教程，综合性强，内容涵盖：估计；检验；大样本理论，这些都是研究生要进入博士或者更高层次必须学习的预备知识。为了让读者具备更加强硬的数学背景和更广阔的理论知识，书中不仅给出了经典方法，也给出了贝叶斯推理知识。目次：概率模型；充分统计量；决策理论；假设检验；估计；等价；大样本理论；分层模型；序列分析；附录：测度与积分理论；概率论；数学定理；分布概述。
　　《数学与金融经典教材：统计理论（影印版）》读者对象：概率统计、数学专业以及相关专业的高年级本科生、研究生和相关的科研人员。

After a brief review of elementary statistical theory， the coverage of thesubject matter begins with a detailed treatment of parametric statisticalmodels as motivated by DeFinetti's representation theorem for exchangeablerandom variables （Chapter l）. In addition， Dirichlet processes and othertailfree processes are presented as examples of infinite-dimensional param-eters. Chapter 2 introduces sufficient statistics from both Bayesian andnon-Bayesian viewpoints. Exponential families are discussed here becauseof the important role sufficiency plays in these models. Also， the conceptof iriformation is introduced together with its relationship to sufficiency.A representation theorem is given for general distributions based on suffi-cient statistics. Decision theory is the subject of Chapter 3， which includesdiscussions of admissibility and minimaxity. Section 3.3 presents an ax-iomatic derivation of Bayesian decision theory， including the use of condi-tional probability. Chapter 4 covers hypothesis testing， including unbiasedtests， P-values， and Bayes factors. We highlight the contrasts between thetraditional "uniformly most powerful" （UMP） approach to testing and de-cision theoretic approaches （both Bayesian and classical）. In particular， wesee how the asymmetric treatment of hypotheses and alternatives in theUMP approach accounts for much of the difference. Point and set estima-tion are the topics of Chapter 5. This includes unbiased and maximum like-lihood estimation as well as confidence， prediction， and tolerance sets. Wealso introduce robust estimation and the bootstrap. Equivariant decisionrules are covered in Chapter 6. In Section 6.2.2， we debunk the commonmisconception of equivariant rules as means for preserving decisions un-der changes of measurement scale. Large sample theory is the subject ofChapter 7. This includes asymptotic properties of sample quantiles， maxi-mum likelihood estimators， robust estimators， and posterior distributions.The last two chapters cover situations in which the random variables are
　　not modeled as being exchangeable. Hierarchical models （Chapter 8） areuseful for data arrays. Here， the parameters of the model can be modeled as exchangeable while the observables are only partially exchangeable. We introduce the popular computational tool known as Markov chain MonteOarlo， Gibbs sampling， or successive substitution sampling， which is very useful for fitting hierarchical models. Some topics in sequential analysis are presented in Chapter 9. These include classical tests， Bayesian decisions， confidence sets， and the issue of sampling to a foregone conclusion.