Theoretical Statistics

A text that stresses the general concepts of the theory of statistics Theoretical Statistics provides a systematic statement of the theory of statistics, emphasizing general concepts rather than mathematical rigor. Chapters 1 through 3 provide an overview of statistics and discuss some of the basic philosophical ideas and problems behind statistical procedures. Chapters 4 and 5 cover hypothesis testing with simple and null hypotheses, respectively. Subsequent chapters discuss non-parametrics, interval estimation, point estimation, asymptotics, Bayesian procedure, and deviation theory. Student familiarity with standard statistical techniques is assumed.

Preface

xi

Introduction

1

(10)

Objectives of statistical analysis and theory

1

(4)

Criteria for the choice of families of models

5

(2)

The analysis of complex responses

7

(1)

Plan of the book

8

(3)

Bibliographic notes

10

(1)

Some general concepts

11

(53)

The likelihood

11

(7)

Sufficient statistics

18

(18)

Some general principles of statistical inference

36

(9)

Some approaches to statistical inference

45

(19)

Bibliographic notes

56

(2)

Further results and exercises

58

(6)

Pure significance tests

64

(24)

Null hypotheses

64

(2)

Test statistics and their null distributions

66

(7)

Composite null hypotheses

73

(3)

Discussion

76

(12)

Bibliographic notes

82

(1)

Further results and exercises

83

(5)

Significance tests: simple null hypotheses

88

(43)

General

88

(2)

Formulation

90

(1)

Simple null hypothesis and simple alternative hypothesis

91

(2)

Some examples

93

(6)

Discrete problems

99

(2)

Composite alternatives

101

(4)

Two-sided tests

105

(1)

Local power

106

(15)

Multidimensional alternatives

121

(10)

Bibliographic notes

125

(1)

Further results and exercises

126

(5)

Significance tests: composite null hypotheses

131

(48)

General

131

(3)

Similar regions

134

(23)

Invariant tests

157

(14)

Some more difficult problems

171

(8)

Bibliographic notes

174

(1)

Further results and exercises

175

(4)

Distribution-free and randomization tests

179

(28)

General

179

(3)

Permutation tests

182

(5)

Rank tests

187

(9)

Randomization tests

196

(2)

Distance tests

198

(9)

Bibliographic notes

202

(1)

Further results and exercises

203

(4)

Interval estimation

207

(43)

Introduction

207

(1)

Scalar parameter

208

(20)

Scalar parameter with nuisance parameters

228

(8)

Vector parameter

236

(6)

Estimation of future observations

242

(8)

Bibliographic notes

246

(1)

Further results and exercises

247

(3)

Point estimation

250

(29)

General

250

(2)

General considerations on bias and variance

252

(2)

Cramer-Rao inequality

254

(4)

Achievement of minimum variance and removal of bias

258

(8)

Estimates of minimum mean squared error

266

(4)

Robust estimation

270

(9)

Bibliographic notes

272

(1)

Further results and exercises

273

(6)

Asymptotic theory

279

(85)

Introduction

279

(4)

Maximum likelihood estimates

283

(28)

Large-sample parametric significance tests

311

(33)

Robust inference for location parameters

344

(20)

Bibliographic notes

354

(2)

Further results and exercises

356

(8)

Bayesian methods

364

(48)

Introduction

364

(1)

Bayes's theorem

365

(4)

Conjugate prior distributions

369

(6)

Interpretation of Bayesian probability statements

375

(15)

Bayesian versions of some previous types of procedure

390

(9)

Asymptotic Bayesian theory

399

(1)

Empirical Bayes procedures

400

(12)

Bibliographic notes

406

(1)

Further results and exercises

406

(6)

Decision theory

412

(50)

Examples of statistical decision problems

412

(3)

Formulation of terminal decision problems

415

(2)

Solution of the fully specified terminal decision problem

417

(9)

Utility

Incompletely specified decision problems

426

(3)

Decision theory without prior distributions

429

(12)

Decision theory representation of common problems

441

(4)

A remarkable case of inadmissibility

445

(6)

Sequential decision problems

451

(11)

Bibliographic notes

458

(1)

Further results and exercises

459

(3)

Appendix 1 Determination of probability distributions

462

(4)

Appendix 2 Order statistics

466

(9)

A2.1 General properties

466

(1)

A2.2. Special distributions

467

(1)

A2.3. Asymptotic distributions

468

(2)

A2.4. Linear combinations of order statistics

470

(2)

A2.5. Extreme value theory

472

(3)

Bibliographic notes

474

(1)

Appendix 3 Second-order regression for arbitrary random variables

475

(3)

References

478

(18)

Author Index

496

(3)

Subject Index

499

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Theoretical Statistics

Summary

Table of Contents

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Theoretical Statistics

Summary

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