Information and Exponential Families

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.

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 11.1 Introductory remarks and outline 11.2 Some mathematical prerequisites 21.3 Parametric models 7Part I Lods functions and inferential separationCHAPTER 2 LIKELIHOOD AND PLAUSIBILITY 112.1 Universality 112.2 Likelihood functions and plausibility functions 122.3 Complements 162.4 Notes 16CHAPTER 3 SAMPLE-HYPOTHESIS DUALITY AND LODS FUNCTIONS 193.1 Lods functions 203.2 Prediction functions 233.3 Independence 263.4 Complements 303.5 Notes 31CHAPTER 4 LOGIC OF INFERENTIAL SEPARATION. ANCILLARITY AND SUFFICIENCY 334.1 On inferential separation. Ancillarity and sufficiency 334.2 B-sufficiency and B-ancillarity 384.3 Nonformation 464.4 S-, G-, and M-ancillarity and -sufficiency 494.5 Quasi-ancillarity and Quasi-sufficiency 574.6 Conditional and unconditional plausibility functions 584.7 Complements 624.8 Notes 68Part II Convex analysis, unimodality, and Laplace transformsCHAPTER 5 CONVEX ANALYSIS 735.1 Convex sets 735.2 Convex functions 765.3 Conjugate convex functions 805.4 Differential theory 845.5 Complements 89CHAPTER 6 LOG-CONCAVITY AND UNIMODALITY 936.1 Log-concavity 936.2 Unimodality of continuous-type distributions 966.3 Unimodality of discrete-type distributions 986.4 Complements 100CHAPTER 7 LAPLACE TRANSFORMS 1037.1 The Laplace transform 1037.2 Complements 107Part III Exponential familiesCHAPTER 8 INTRODUCTORY THEORY OF EXPONENTIAL FAMILIES 1118.1 First properties 1118.2 Derived families 1258.3 Complements 1338.4 Notes 136CHAPTER 9 DUALITY AND EXPONENTIAL FAMILIES 1399.1 Convex duality and exponential families 1409.2 Independence and exponential families 1479.3 Likelihood functions for full exponential families 1509.4 Likelihood functions for convex exponential families 1589.5 Probability functions for exponential families 1649.6 Plausibility functions for full exponential families 1689.7 Prediction functions for full exponential families 1709.8 Complements 1739.9 Notes 190CHAPTER 10 INFERENTIAL SEPARATION AND EXPONENTIAL FAMILIES 19110.1 Quasi-ancillarity and exponential families 19110.2 Cuts in general exponential families 19610.3 Cuts in discrete-type exponential families 20210.4 S-ancillarity and exponential families 20810.5 M-ancillarity and exponential families 21110.6 Complement 21810.7 Notes 219References 221Author index 231Subject index 233