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Information and Exponential Families in Statistical Theory
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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

Ole Barndorff-Nielsen is a renowned Danish statistician, Professor Emeritus at Aarhus University at the Thiele Centre for Applied Mathematics in Natural Science and affiliated with the Center for Research in Econometric Analysis of Time Series (CREATES). Since 2008 he has also been affiliated to Institute of Advanced Studies, Technical University Munich.