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Statistical Inference a Work in Progress Ronald Christensen Department of Mathematics and Statistics University of New Mexico Copyright c 2019 Statistical Inference A Work in Progress Springer v Seymour and Wes. Preface “But to us, probability is the very guide of life.” Joseph Butler (1736). The Analogy of Religion, Natural and Revealed, to the Constitution and Course of Nature, Introduction. https://www.loc.gov/resource/dcmsiabooks. analogyofreligio00butl_1/?sp=41 Normally, I wouldn’t put anything this incomplete on the internet but I wanted to make parts of it available to my Advanced Inference Class, and once it is up, you have lost control. Seymour Geisser was a mentor to Wes Johnson and me. He was Wes’s Ph.D. advisor. Near the end of his life, Seymour was finishing his 2005 book Modes of Parametric Statistical Inference and needed some help. Seymour asked Wes and Wes asked me. I had quite a few ideas for the book but then I discovered that Sey- mour hated anyone changing his prose. That was the end of my direct involvement. The first idea for this book was to revisit Seymour’s. (So far, that seems only to occur in Chapter 1.) Thinking about what Seymour was doing was the inspiration for me to think about what I had to say about statistical inference. And much of what I have to say is inspired by Seymour’s work as well as the work of my other professors at Min- nesota, notably Christopher Bingham, R. Dennis Cook, Somesh Das Gupta, Mor- ris L. Eaton, Stephen E. Fienberg, Narish Jain, F. Kinley Larntz, Frank B. Martin, Stephen Orey, William Suderth, and Sanford Weisberg. No one had a greater influ- ence on my career than my advisor, Donald A. Berry. I simply would not be where I am today if Don had not taken me under his wing. The material in this book is what I (try to) cover in the first semester of a one year course on Advanced Statistical Inference. The other semester I use Ferguson (1996). The course is for students who have had at least Advanced Calculus and hopefully some Introduction to Analysis. The course is at least as much about introducing some mathematical rigor into their studies as it is about teaching statistical inference. (Rigor also occurs in our Linear Models class but there it is in linear algebra and vii viii Preface here it is in analysis.) I try to introduce just enough measure theory for students to get an idea of its value (and to facility my presentations). But I get tired of doing analysis, so occasionally I like to teach some inference in the advanced inference course. Many years ago I went to a JSM (Joint Statistical Meeting) and heard Brian Joiner make the point that everybody learns from examples to generalities/theory. Ever since I have tried, with varying amounts of success, to incorporate this dictum into my books. (Plane Answers’ first edition preceded that experience.) This book has four chapters of example based discussion before it begins the theory in Chapter 5. There are only three chapters of theory but they are tied to extensive appendices on technical material. The first appendix merely reviews basic (non-measure theoretic) ideas of multivariate distributions. The second briefly introduces ideas of measure theory, measure theoretic probability, and convergence. Appendix C introduces the measure theoretic approaches to conditional probability and conditional expecta- tion. Appendix D adds a little depth (very little) to the discussion of measure theory and probability. Appendix E introduces the concept of identifiability. Appendix F merely reviews concepts of multivariate differentiation. Chapters 8 through 13 are me being self-indulgent and tossing in things of personal interest to me. (They don’t actually get covered in the class.) References to PA and ALM are to my books Plane Answers and Advanced Linear Modeling. Preface ix Recommended Additional Reading Ferguson, T. S. (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York. I would use this as a text if it were not out of print! I may have borrowed even more than I realized from this. Rao, C. R. (1973). Linear Statistical Inference and Its Applications, Second Edition. John Wiley and Sons, New York. Covers almost everything I want to cover. Not bad to read but I have found it impossible to teach out of. Cramer,´ H. (1946). Mathematical Methods of Statistics. Princeton University Press, Princeton. Excellent! All the analysis you need to do math stat? Limited because of age. Lehmann, E. L. (1983) Theory of Point Estimation. John Wiley and Sons, New York. (Now Lehmann and Casella.) Lehmann, E. L. (1986) Testing Statistical Hypotheses, Second Edition. John Wiley and Sons, New York. (Now Lehmann and Romano.) Berger, J. O. (1993). Statistical Decision Theory and Bayesian Analysis. Revised Second Edition. Springer-Verlag, New York. We used to teach Advanced Inference out of above three books. Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman and Hall, London. More profound on ideas, less on math. Manoukian, E. B. (1986), Modern Concepts and Theorems of Mathematical Statistics. Springer- Verlag, New York. SHORT! Usually the first book off my shelf. Statements, not proofs or explanations. Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, Third Edition. John Wiley and Sons, New York. For our purposes, a source for multivariate normal only. An alternative to PA for this. Wilks, Mathematical Statistics; Zacks, Theory of Statistical Inference. Both old but thorough. Wilks is great for order statistics and distributions related to discrete data. Wasserman, Larry (2004). All of Statistics. Springer, New York. Statistical Inference Cox, D.R. (2006). Principles of Statistical Inference. Cambridge University Press, Cambridge. Fisher, R.A. (1956). Statistical Methods and Scientific Inference, Third Edition, 1973. Hafner Press, New York. Geisser, S. (1993). Modes of Parametric Statistical Inference. Wiley, New York. Bayesian Books de Finetti, B. (1974, 1975). Theory of Probability, Vols. 1 and 2. John Wiley and Sons, New York. Jeffreys, H. (1961). Theory of Probability, Third Edition. Oxford University Press, London. Savage, L. J. (1954). The Foundations of Statistics. John Wiley and Sons, New York. DeGroot, M. H. (1970). Optimal Statistical Decisions. McGraw-Hill, New York. First three are foundational. There are now TONS of other books, see mine for other references. Large Sample Theory Books Ferguson, Thomas S. (1996). A Course in Large Sample Theory. Chapman and Hall, New York. I teach out of this. (Need I say more?) Lehmann, E. L. (1999) Elements of Large-Sample Theory. Springer. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, (paperback, 2001) I haven’t read either of the last two, but I hear they are good. x Preface Probability and Measure Theory Books Ash, Robert B. and Doleans-Dade, Catherine A. (2000). Probability and Measure Theory, Second Edition. Academic Press, San Diego. I studied the first edition of this while in grad school and continue to use it as a reference. Billingsley, Patrick (2012). Probability and Measure, Fourth Edition. Wiley, New York. I haven’t read this, but I hear it is good. There are lots of others. There are also lots of good books on probability alone. Contents Preface ............................................................ vii Table of Contents ................................................... ix 1 Overview ...................................................... 1 1.1 Early Examples............................................1 1.2 Testing...................................................1 1.3 Decision Theory...........................................2 1.4 Some Ideas about Inference..................................2 1.5 The End..................................................2 1.6 The Bitter End.............................................3 2 Significance Tests .............................................. 5 2.1 Generalities...............................................5 2.2 Continuous Distributions.................................... 13 2.2.1 One Sample Normals................................. 13 2.3 Testing Two Sample Variances................................ 19 2.4 Fisher’s z distribution........................................ 21 2.5 Final Notes................................................ 27 3 Hypothesis Tests ............................................... 29 3.1 Testing Two Simple Hypotheses.............................. 29 3.1.1 Neyman-Pearson tests................................ 30 3.1.2 Bayesian Tests...................................... 33 3.2 Simple Versus Composite Hypotheses......................... 34 3.2.1 Neyman-Pearson Testing.............................. 35 3.2.2 Bayesian Testing..................................... 36 3.3 Composite versus Composite................................. 37 3.3.1 Neyman-Pearson Testing.............................. 37 3.3.2 Bayesian Testing..................................... 38 3.4 More on Neyman-Pearson Tests.............................. 38 xiii xiv Contents 3.5 More on Bayesian Testing................................... 39 3.6 Hypothesis Test P Values.................................... 40 3.7 Permutation Tests.......................................... 41 4 Comparing Testing Procedures .................................. 43 4.1 Discussion................................................ 44 4.2 Jeffreys’ Critique........................................... 46 5 Decision Theory................................................ 47 5.1 Optimal Prior Actions......................................
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