i i “Schweder-Book” — 2015/10/21 — 17:46 — page i — #1 i i Confidence, Likelihood, Probability This lively book lays out a methodology of confidence distributions and puts them through their paces. Among other merits, they lead to optimal combinations of confidence from different sources of information, and they can make complex models amenable to objective and indeed prior-free analysis for less subjectively inclined statisticians. The generous mixture of theory, illustrations, applications and exercises is suitable for statisticians at all levels of experience, as well as for data-oriented scientists. Some confidence distributions are less dispersed than their competitors. This concept leads to a theory of risk functions and comparisons for distributions of confidence. Neyman–Pearson type theorems leading to optimal confidence are developed and richly illustrated. Exact and optimal confidence distributions are the gold standard for inferred epistemic distributions in empirical sciences. Confidence distributions and likelihood functions are intertwined, allowing prior distributions to be made part of the likelihood. Meta-analysis in likelihood terms is developed and taken beyond traditional methods, suiting it in particular to combining information across diverse data sources. TORE SCHWEDER is a professor of statistics in the Department of Economics and at the Centre for Ecology and Evolutionary Synthesis at the University of Oslo. NILS LID HJORT is a professor of mathematical statistics in the Department of Mathematics at the University of Oslo. i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page ii — #2 i i CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS Editorial Board Z. Ghahramani (Department of Engineering, University of Cambridge) R. Gill (Mathematical Institute, Leiden University) F. P. Kelly (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge) B. D. Ripley (Department of Statistics, University of Oxford) S. Ross (Department of Industrial and Systems Engineering, University of Southern California) M. Stein (Department of Statistics, University of Chicago) This series of high-quality upper-division textbooks and expository monographs covers all aspects of stochastic applicable mathematics. The topics range from pure and applied statistics to probability theory, operations research, optimization and mathematical programming. The books contain clear presentations of new developments in the field and also of the state of the art in classical methods. While emphasizing rigorous treatment of theoretical methods, the books also contain applications and discussions of new techniques made possible by advances in computational practice. A complete list of books in the series can be found at www.cambridge.org/statistics. Recent titles include the following: 15. Measure Theory and Filtering, by Lakhdar Aggoun and Robert Elliott 16. Essentials of Statistical Inference, by G. A. Young and R. L. Smith 17. Elements of Distribution Theory, by Thomas A. Severini 18. Statistical Mechanics of Disordered Systems, by Anton Bovier 19. The Coordinate-Free Approach to Linear Models, by Michael J. Wichura 20. Random Graph Dynamics,byRickDurrett 21. Networks, by Peter Whittle 22. Saddlepoint Approximations with Applications, by Ronald W. Butler 23. Applied Asymptotics, by A. R. Brazzale, A. C. Davison and N. Reid 24. Random Networks for Communication, by Massimo Franceschetti and Ronald Meester 25. Design of Comparative Experiments, by R. A. Bailey 26. Symmetry Studies,byMarlosA.G.Viana 27. Model Selection and Model Averaging, by Gerda Claeskens and Nils Lid Hjort 28. Bayesian Nonparametrics, edited by Nils Lid Hjort et al. 29. From Finite Sample to Asymptotic Methods in Statistics, by Pranab K. Sen, Julio M. Singer and Antonio C. Pedrosa de Lima 30. Brownian Motion, by Peter Morters¨ and Yuval Peres 31. Probability (Fourth Edition), by Rick Durrett 33. Stochastic Processes,byRichardF.Bass 34. Regression for Categorical Data,byGerhardTutz 35. Exercises in Probability (Second Edition),byLo¨ıc Chaumont and Marc Yor 36. Statistical Principles for the Design of Experiments, by R. Mead, S. G. Gilmour and A. Mead 37. Quantum Stochastics, by Mou-Hsiung Chang 38. Nonparametric Estimation under Shape Constraints, by Piet Groeneboom and Geurt Jungbloed 39. Large Sample Covariance Matrices, by Jianfeng Yao, Zhidong Bai and Shurong Zheng 40. Mathematical Foundations of Infinite-Dimensional Statistical Models,byEvaristGineand´ Richard Nickl i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page iii — #3 i i Confidence, Likelihood, Probability Statistical Inference with Confidence Distributions Tore Schweder University of Oslo Nils Lid Hjort University of Oslo i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page iv — #4 i i 32 Avenue of the Americas, New York, NY 10013-2473, USA Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521861601 c Tore Schweder and Nils Lid Hjort 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Schweder, Tore. Confidence, likelihood, probability : statistical inference with confidence distributions / Tore Schweder, University of Oslo, Nils Lid Hjort, University of Oslo. pages cm. – (Cambridge series in statistical and probabilistic mathematics) Includes bibliographical references and index. ISBN 978-0-521-86160-1 (hardback : alk. paper) 1. Observed confidence levels (Statistics) 2. Mathematical statistics. 3. Probabilities. I. Hjort, Nils Lid. II. Title. QA277.5.S39 2016 519.2–dc23 2015016878 ISBN 978-0-521-86160-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page v — #5 i i To my four children –T.S. To my four brothers –N.L.H. i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page vi — #6 i i i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page vii — #7 i i Contents Preface page xiii 1 Confidence, likelihood, probability: An invitation 1 1.1 Introduction 1 1.2 Probability 4 1.3 Inverse probability 6 1.4 Likelihood 7 1.5 Frequentism 8 1.6 Confidence and confidence curves 10 1.7 Fiducial probability and confidence 14 1.8 Why not go Bayesian? 16 1.9 Notes on the literature 19 2 Inference in parametric models 23 2.1 Introduction 23 2.2 Likelihood methods and first-order large-sample theory 24 2.3 Sufficiency and the likelihood principle 30 2.4 Focus parameters, pivots and profile likelihoods 32 2.5 Bayesian inference 40 2.6 Related themes and issues 42 2.7 Notes on the literature 48 Exercises 50 3 Confidence distributions 55 3.1 Introduction 55 3.2 Confidence distributions and statistical inference 56 3.3 Graphical focus summaries 65 3.4 General likelihood-based recipes 69 3.5 Confidence distributions for the linear regression model 72 3.6 Contingency tables 78 3.7 Testing hypotheses via confidence for alternatives 80 3.8 Confidence for discrete parameters 83 vii i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page viii — #8 i i viii Contents 3.9 Notes on the literature 91 Exercises 92 4 Further developments for confidence distribution 100 4.1 Introduction 100 4.2 Bounded parameters and bounded confidence 100 4.3 Random and mixed effects models 107 4.4 The Neyman–Scott problem 111 4.5 Multimodality 115 4.6 Ratio of two normal means 117 4.7 Hazard rate models 122 4.8 Confidence inference for Markov chains 128 4.9 Time series and models with dependence 133 4.10 Bivariate distributions and the average confidence density 138 4.11 Deviance intervals versus minimum length intervals 140 4.12 Notes on the literature 142 Exercises 144 5 Invariance, sufficiency and optimality for confidence distributions 154 5.1 Confidence power 154 5.2 Invariance for confidence distributions 157 5.3 Loss and risk functions for confidence distributions 161 5.4 Sufficiency and risk for confidence distributions 165 5.5 Uniformly optimal confidence for exponential families 173 5.6 Optimality of component confidence distributions 177 5.7 Notes on the literature 179 Exercises 180 6 The fiducial argument 185 6.1 The initial argument 185 6.2 The controversy 188 6.3 Paradoxes 191 6.4 Fiducial distributions and Bayesian posteriors 193 6.5 Coherence by restricting the range: Invariance or irrelevance? 194 6.6 Generalised fiducial inference 197 6.7 Further remarks 200 6.8 Notes on the literature 201 Exercises 202 7 Improved approximations for confidence distributions 204 7.1 Introduction 204 7.2 From first-order to second-order approximations 205 i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page ix — #9 i i Contents ix 7.3 Pivot tuning 208 7.4 Bartlett corrections for the deviance 210 7.5 Median-bias correction 214 7.6 The t-bootstrap and abc-bootstrap method 217 7.7 Saddlepoint approximations and