# Information and Exponential Families: In Statistical Theory

ISBN: 978-1-118-85728-1 May 2014 248 Pages

## Description

First published by Wiley in 1978, this book is being re-issued with a new Preface by the author. The roots of the book lie in the writings of RA Fisher both as concerns results and the general stance to statistical science, and this stance was the determining factor in the author's selection of topics. His treatise brings together results on aspects of statistical information, notably concerning likelihood functions, plausibility functions, ancillarity, and sufficiency, and on exponential families of probability distributions.
CHAPTER 1 INTRODUCTION 1

1.1 Introductory remarks and outline 1

1.2 Some mathematical prerequisites 2

1.3 Parametric models 7

Part I Lods functions and inferential separation

CHAPTER 2 LIKELIHOOD AND PLAUSIBILITY 11

2.1 Universality 11

2.2 Likelihood functions and plausibility functions 12

2.3 Complements 16

2.4 Notes 16

CHAPTER 3 SAMPLE-HYPOTHESIS DUALITY AND LODS FUNCTIONS 19

3.1 Lods functions 20

3.2 Prediction functions 23

3.3 Independence 26

3.4 Complements 30

3.5 Notes 31

CHAPTER 4 LOGIC OF INFERENTIAL SEPARATION. ANCILLARITY AND SUFFICIENCY 33

4.1 On inferential separation. Ancillarity and sufficiency 33

4.2 B-sufficiency and B-ancillarity 38

4.3 Nonformation 46

4.4 S-, G-, and M-ancillarity and -sufficiency 49

4.5 Quasi-ancillarity and Quasi-sufficiency 57

4.6 Conditional and unconditional plausibility functions 58

4.7 Complements 62

4.8 Notes 68

Part II Convex analysis, unimodality, and Laplace transforms

CHAPTER 5 CONVEX ANALYSIS 73

5.1 Convex sets 73

5.2 Convex functions 76

5.3 Conjugate convex functions 80

5.4 Differential theory 84

5.5 Complements 89

CHAPTER 6 LOG-CONCAVITY AND UNIMODALITY 93

6.1 Log-concavity 93

6.2 Unimodality of continuous-type distributions 96

6.3 Unimodality of discrete-type distributions 98

6.4 Complements 100

CHAPTER 7 LAPLACE TRANSFORMS 103

7.1 The Laplace transform 103

7.2 Complements 107

Part III Exponential families

CHAPTER 8 INTRODUCTORY THEORY OF EXPONENTIAL FAMILIES 111

8.1 First properties 111

8.2 Derived families 125

8.3 Complements 133

8.4 Notes 136

CHAPTER 9 DUALITY AND EXPONENTIAL FAMILIES 139

9.1 Convex duality and exponential families 140

9.2 Independence and exponential families 147

9.3 Likelihood functions for full exponential families 150

9.4 Likelihood functions for convex exponential families 158

9.5 Probability functions for exponential families 164

9.6 Plausibility functions for full exponential families 168

9.7 Prediction functions for full exponential families 170

9.8 Complements 173

9.9 Notes 190

CHAPTER 10 INFERENTIAL SEPARATION AND EXPONENTIAL FAMILIES 191

10.1 Quasi-ancillarity and exponential families 191

10.2 Cuts in general exponential families 196

10.3 Cuts in discrete-type exponential families 202

10.4 S-ancillarity and exponential families 208

10.5 M-ancillarity and exponential families 211

10.6 Complement 218

10.7 Notes 219

References 221

Author index 231

Subject index 233