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NOTES FOR QUANTITATIVE METHODS: SOME ADVANCED MATHEMATICS FROM AN ELEMENTARY POINT OF VIEW MAXWELL B. STINCHCOMBE Contents 0. Organizational Stuff 1. Some Basics About Numbers and Quantities 1.1. Lengths and measurements 1.2. Why we want more 1.3. Valuing sequences of rewards 1.4. Convex analysis 1.5. Problems 1.6. Self-guided tour to differentiability and concavity 2. Some Basic Results in Metric Spaces 2.1. Metrics 2.2. Probability distributions as cdf’s 2.3. Continuity 2.4. Compactness and the existence of optima 2.5. The Theorem of the Maximum 2.6. The Separating Hyperplane Theorem 2.7. Problems 3. Dynamic Programming, Deterministic and Stochastic 3.1. Compactness and continuity in spaces of sequences 3.2. Deterministic Dynamic Programming 3.3. Stochastic Dynamic Programming 3.4. Problems 4. An Overview of the Statistics Part of this Course 4.1. Basics 4.2. Other properties of estimators 4.3. Bayesians 4.4. Classical statistics Date: November 25, 2003. Fall Semester, 2003. Unique #30238. 1 4 5 5 5 5 6 9 11 18 18 18 19 19 19 20 20 22 22 22 24 25 27 27 27 28 29

2 MAXWELL B. STINCHCOMBE 4.5. An Information Inequality 4.6. Mis-Specification 4.7. Problems 5. Basic Probability, Transformations, and Expectations 5.1. Basic Probability and Expectations 5.2. Transformations and Expectations 5.3. Problems 6. Some Continuous Distributions 6.1. Uniform distributions, U[θ1 , θ2 ] 6.2. The normal or Gaussian family of distributions, N(µ, σ 2 ) 6.3. A useful device 6.4. The gamma family, Γ(α, β) 6.5. Special cases of Γ(α, β) distributions 6.6. Cauchy random variables 6.7. Exponential Families 6.8. Some (in)equalities 6.9. Problems 7. Random Vectors, Conditional Expectations, Independence 7.1. Dependence, conditional probabilities and expectations 7.2. Projections 7.3. Causality and conditional probability 7.4. Independence, sums of independent rv’s 7.5. Covariance and correlation 7.6. Bivariate normals 7.7. A pair of discrete, portfolio management examples 7.8. The matrix formulation 7.9. Problems 8. Sampling Distributions and Normal Approximations 9. Sufficient Statistics as Data Compression 9.1. Sufficient statistics 9.2. Rao-Blackwell 9.3. Problems 10. Finding and Evaluating Estimators 10.1. The basic Gaussian example 10.2. Some examples of finding estimators 10.3. Problems 11. Evaluating different estimators 30 31 32 34 34 34 34 37 37 38 39 39 41 41 42 42 42 44 44 44 45 46 46 47 47 50 51 52 54 54 55 56 57 57 58 60 61

QUANTITATIVE METHODS 11.1. Mean Squared Error (MSE) 11.2. Desirable properties for estimators 11.3. The Cram´er-Rao lower bound 11.4. Problems 12. Hypothesis Testing 12.1. Overview 12.2. The perfect power function and types of errors 12.3. Some generalities about the probabilities of the different types of errors 12.4. The Likelihood Ratio Tests 12.5. Confidence intervals, p-values, and hypothesis testing 12.6. Problems 3 61 62 62 62 64 64 64 65 66 67 68

4 MAXWELL B. STINCHCOMBE 0. Organizational Stuff Meetings: Mondays and Wednesdays, 2-3:30 and Wednesdays 8:30-9:30, in BRB 1.120. Teachers: My office is BRB 2.118, phone number is 475-8515, e-mail address is maxwell@eco.utexas.edu office hours Mondays and Wednesdays 10-12. You are very lucky to have Lori Stuntz as the T.A. for this course. Her office is 4.116, e-mail address is stuntz@eco.utexas.edu, office hours TBA. Texts: For the statistical part of the course, we’ll use George Casella and Roger Berger’s Statistical Inference, 2’nd ed. (Duxbury 2002), following it fairly closely. For the optimization and analysis parts of the course, we’ll use Sheldon M. Ross’s Applied Probability Models with Optimization Applications (Dover Publications, 1992) and A. N. Kolmogorov and S. V. Fomin’s Introductory Real Analysis (Dover Publications, 1970). Throughout, you will be refering to the microeconomics textbook, Microeconomic Theory, by Mas-Colell, Whinston, and Green. Topics: Completeness properties of R and R` , summability and valuation of streams of utilities, convex analysis and duality; further properties of R` and related spaces, (including compactness, continuity and measurability of functions on R` , summability of sequences, existence of optima, fixed point theorems, cdf’s, other metrics, other metric spaces, the Theorem of the Maximum); Probabilities and expectations (including domains, modes of convergence, convergence theorems, orders of stochastic dominance, conditional expectations and probabilities); Dynamic programming (including properties of sequence spaces and probabilities on them, Bellman and Euler equations, the role of the Theorem of the Maximum, growth models); Statistics (including specific distributions [uniform, gamma, beta, Gaussian, t, F , χ2 , Poisson, negative exponential, Weibull, logistic], estimators and their properties [consistency, Glivenko-Cantelli, different kinds of “best” estimators, Bayesian estimators, MLE estimators, information inequalities, sufficiency, Blackwell-Rao], properties of hypothesis tests [types of errors and their associated distributions, the Neyman-Pearson Lemma]).

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