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Lecture Notes on Advanced Statistical Theory1

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Lecture Notes on Advanced Statistical Theory1 Lecture Notes on Advanced Statistical Theory1 Ryan Martin Department of Statistics North Carolina State University www4.stat.ncsu.edu/~rmartin January 3, 2017 1These notes were written to supplement the lectures for the Stat 511 course given by the author at the University of Illinois Chicago. The accompanying textbook for the course is Keener's Theoretical Statistics, Springer, 2010, and is referred to frequently though out the notes. The author makes no guarantees that these notes are free of typos or other, more serious errors. Feel free to contact the author if you have any questions or comments. Contents 1 Introduction and Preparations 4 1.1 Introduction . .4 1.2 Mathematical preliminaries . .5 1.2.1 Measure and integration . .5 1.2.2 Basic group theory . 10 1.2.3 Convex sets and functions . 11 1.3 Probability . 12 1.3.1 Measure-theoretic formulation . 12 1.3.2 Conditional distributions . 14 1.3.3 Jensen's inequality . 16 1.3.4 A concentration inequality . 17 1.3.5 The \fundamental theorem of statistics" . 19 1.3.6 Parametric families of distributions . 21 1.4 Conceptual preliminaries . 22 1.4.1 Ingredients of a statistical inference problem . 22 1.4.2 Reasoning from sample to population . 23 1.5 Exercises . 25 2 Exponential Families, Sufficiency, and Information 29 2.1 Introduction . 29 2.2 Exponential families of distributions . 30 2.3 Sufficient statistics . 33 2.3.1 Definition and the factorization theorem . 33 2.3.2 Minimal sufficient statistics . 36 2.3.3 Ancillary and complete statistics . 38 2.4 Fisher information . 41 2.4.1 Definition . 41 2.4.2 Sufficiency and information . 42 2.4.3 Cramer{Rao inequality . 43 2.4.4 Other measures of information . 44 2.5 Conditioning . 45 2.6 Discussion . 47 2.6.1 Generalized linear models . 47 1 2.6.2 A bit more about conditioning . 48 2.7 Exercises . 49 3 Likelihood and Likelihood-based Methods 54 3.1 Introduction . 54 3.2 Likelihood function . 54 3.3 Likelihood-based methods and first-order theory . 55 3.3.1 Maximum likelihood estimation . 55 3.3.2 Likelihood ratio tests . 58 3.4 Cautions concerning the first-order theory . 60 3.5 Alternatives to the first-order theory . 62 3.5.1 Bootstrap . 62 3.5.2 Monte Carlo and plausibility functions . 63 3.6 On advanced likelihood theory . 63 3.6.1 Overview . 63 3.6.2 \Modified” likelihood . 64 3.6.3 Asymptotic expansions . 66 3.7 A bit about computation . 67 3.7.1 Optimization . 67 3.7.2 Monte Carlo integration . 67 3.8 Discussion . 68 3.9 Exercises . 69 4 Bayesian Inference 75 4.1 Introduction . 75 4.2 Bayesian analysis . 77 4.2.1 Basic setup of a Bayesian inference problem . 77 4.2.2 Bayes's theorem . 77 4.2.3 Inference . 79 4.2.4 Marginalization . 80 4.3 Some examples . 81 4.4 Motivations for the Bayesian approach . 84 4.4.1 Some miscellaneous motivations . 84 4.4.2 Exchangeability and deFinetti's theorem . 85 4.5 Choice of priors . 88 4.5.1 Prior elicitation . 88 4.5.2 Convenient priors . 88 4.5.3 Many candidate priors and robust Bayes . 88 4.5.4 Objective or non-informative priors . 89 4.6 Bayesian large-sample theory . 90 4.6.1 Setup . 90 4.6.2 Laplace approximation . 91 4.6.3 Bernstein{von Mises theorem . 92 2 4.7 Concluding remarks . 94 4.7.1 Lots more details on Bayesian inference . 94 4.7.2 On Bayes and the likelihood principle . 94 4.7.3 On the \Bayesian" label . 95 4.7.4 On \objectivity" . 95 4.7.5 On the role of probability in statistical inference . 96 4.8 Exercises . 97 5 Statistical Decision Theory 101 5.1 Introduction . 101 5.2 Admissibility . 103 5.3 Minimizing a \global" measure of risk . 105 5.3.1 Minimizing average risk . 105 5.3.2 Minimizing maximum risk . 109 5.4 Minimizing risk under constraints . 112 5.4.1 Unbiasedness constraints . 112 5.4.2 Equivariance constraints . 113 5.4.3 Type I error constraints . 115 5.5 Complete class theorems . 116 5.6 On minimax estimation of a normal mean . 117 5.7 Exercises . 119 6 More Asymptotic Theory (incomplete!) 122 6.1 Introduction . 122 6.2 M- and Z-estimators . 123 6.2.1 Definition and examples . 123 6.2.2 Consistency . 124 6.2.3 Rates of convergence . 127 6.2.4 Asymptotic normality . 128 6.3 More on asymptotic normality and optimality . 131 6.3.1 Introduction . 131 6.3.2 Hodges's provocative example . 131 6.3.3 Differentiability in quadratic mean . 131 6.3.4 Contiguity . 135 6.3.5 Local asymptotic normality . 135 6.3.6 On asymptotic optimality . 135 6.4 More Bayesian asymptotics . 135 6.4.1 Consistency . 135 6.4.2 Convergence rates . 138 6.4.3 Bernstein{von Mises theorem, revisited . 140 6.5 Concluding remarks . 140 6.6 Exercises . 140 3 Chapter 1 Introduction and Preparations 1.1 Introduction Stat 511 is a first course in advanced statistical theory. This first chapter is intended to set the stage for the material that is the core of the course. In particular, these notes define the notation we shall use throughout, and also set the conceptual and mathematical level we will be working at. Naturally, both the conceptual and mathematical level will be higher than in an intermediate course, such as Stat 411 at UIC. On.
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