Statistical Information and Likelihood

It is an honor to be asked to write a foreword to this book, for I believe that it and other books to follow will eventually lead to a dramatic change in the current statistics curriculum in our universities. I spent the 1975-76 academic year at Florida State University in Tallahassee. My purpose was to complete a book on Statistical Reliability Theory with Frank Proschan. At the time, I was working on total time on test processes. At the same time, I started attending lectures by Dev Basu on statistical inference. It was Lehmann's hypothesis testing course and Lehmann's book was the text. However, I noticed something strange - Basu never opened the book. He was obviously not following it. Instead, he was giving a very elegant, measure theoretic treatment of the concepts of sufficiency, ancillarity, and invariance. He was interested in the concept of information - what it meant. - how it fitted in with contemporary statistics. As he looked at the fundamental ideas, the logic behind their use seemed to evaporate. I was shocked. I didn't like priors. I didn't like Bayesian statistics. But after the smoke had cleared, that was all that was left. Basu loves counterexamples. He is like an art critic in the field of statistical inference. He would find a counterexample to the Bayesian approach if he could. So far, he has failed in this respect.

I Information and Likelihood.- I Recovery of Ancillary Information.- 0. Notes.- 1. Introduction.- 2. The Sample Size Analogy.- 3. A Logical Difficulty.- 4. Conceptual Statistical Experiments.- II Statistical Information and Likelihood Part I: Principles.- 0. Notes.- 1. Statistical Information.- 2. Basic Definitions and Relations.- 3. Some Principles of Inference.- 4. Information as a Function.- 5. Fisher Information.- 6. The Likelihood Principle.- III Statistical Information and Likelihood Part II: Methods.- 0. Notes.- 1. Non-Bayesian Likelihood Methods.- 2. Likelihood: Point Function or a Measure ?.- 3. Maximum Likelihood.- IV Statistical Information and Likelihood Part III: Paradoxes.- 0. Notes.- 1. A Fallacy of Five Terms.- 2. The Stopping Rule Paradox.- 3. The Stein Paradox.- V Statistical Information and Likelihood: Discussions.- 0. Notes.- 1. Discussions.- 2. Barnard-Basu Correspondence.- VI Partial Sufficiency.- 0. Notes.- 1. Introduction.- 2. Specific Sufficient Statistics.- 3. Partial Sufficiency.- 4. H-sufficiency.- 5. Invariantly Sufficient Statistics.- 6. Final Remarks.- VII Elimination of Nuisance Parameters.- 0. Notes.- 1. The Elimination Problem and Methods.- 2. Marginalization and Conditioning.- 3. Partial Sufficiency and Partial Ancillarity.- 4. Generalized Sufficiency and Conditionality Principles.- 5. A Choice Dilemma.- 6. A Conflict.- 7. Rao-Blackwell Type Theorems.- 8. The Bayesian Way.- 9. Unrelated Parameters.- VIII Sufficiency and Invariance.- 0. Notes.- 1. Summary.- 2. Definitions and Preliminaries.- 3. A Mathematical Introduction.- 4. Statistical Motivation.- 5. When a Boundedly Complete Sufficient Sub-field Exists.- 6. The Dominated Case.- 7. Examples.- 8. Transformations of a Set of Normal Variables.- 9. Parameter-preserving Transformations.- 10. Some Typical Invariance Reductions.- 11. Some Final Remarks.- IX Ancillary Statistics, Pivotal Quantities and Confidence.- Statements.- 1. Introduction.- 2. Ancillary Statistics.- 3. Ancillary Information.- 4. Pivotal Quantities.- 5. Confidence Statements.- 6. Ancillarity in Survey Sampling.- II Survey Sampling and Randomization.- X Sufficiency in Survey Sampling.- 1. Introduction and Summary.- 2. Sufficient Statistics and Sub-Fields.- 3. Pitcher and Burkholder Pathologies.- 4. Sufficiency in Typical Sampling Situations.- XI Likelihood Principle and Survey Sampling.- 0. Notes.- 1. Introduction.- 2. Statistical Models and Sufficiency.- 3. Sufficiency in Discrete Models.- 4. The Sample Survey Models.- 5. The Sufficiency and Likelihood Principles.- 6. Role and Choice of the Sampling Plan.- 7. Concluding Remarks.- XII On the Logical Foundations of Survey Sampling.- 1. An Idealization of the Survey Set-up.- 2. Probability in Survey Theory.- 3. Non-sequential Sampling Plans and Unbiased Estimation.- 4. The Label-set and the Sample Core.- 5. Linear Estimation in Survey Sampling.- 6. Homogeneity, Necessary Bestness and Hyper-Admissibility.- 7. Linear Invariance.- XIII On the Logical Foundations of Survey Sampling: Discussions.- 1. Discussions.- 2. Author’s Reply.- XIV Relevance of Randomization in Data Analysis.- 0. Notes.- 1. Introduction.- 2. Likelihood.- 3. A Survey Sampling Model.- 4. Why Randomize?.- 5. Randomization Analysis of Data.- 6. Randomization and Information.- 7. Information in Data.- 8. A Critical Review.- XV The Fisher Randomization Test.- 0. Notes.- 1. Introduction.- 2. Randomization.- 3. Two Fisher Principles.- 4. The Fisher Randomization Test.- 5. Did Fisher Change His Mind?.- 6. Randomization and Paired Comparisons.- 7. Concluding Remarks.- XVI The Fisher Randomization Test: Discussions.- 1. Discussions.- 2. Rejoinder.- III Miscellaneous Notes and Discussions.- XVII Likelihood and Partial Likelihood.- XVIII A Discussion on the Fisher Exact Test.- XIX A Discussion on Survey Theory.- XX A Note on Unbiased Estimation.- XXI The Concept of Asymptotic Efficiency.- XXII Statistics Independent of a Complete Sufficient Statistic.- XXIII Statistics Independent of a Sufficient Statistic.- XXIV The Basu Theorems.- References.