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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.
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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
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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
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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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