Mathematical Statistics
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Tentative Course Schedule
PART 1: Background
1
Probability
1.1
Review
1.1.1
Probability Space
1.1.2
Random Variables
1.1.3
Joint distribution of RVs
1.1.4
Some important random variables
1.1.5
Independent random variables
1.1.6
Transformations of RVs
1.1.7
Expectation
1.2
Moment Generating and Characteristic Functions
1.2.1
Moment Generating Functions
1.2.2
Characteristic Functions
1.3
Inequalities
1.3.1
Typical tail bound inequalities
1.3.2
Exponential concentration inequalities
1.3.3
Inequalities for expectations
1.4
Law of Large Numbers
1.5
Central Limit Theorem
PART 2: Inference
2
Sampling, Estimating CDF and Statistical Functionals
2.1
Sampling
2.1.1
Simple Random Sample
2.1.2
Standard Random Sample
2.2
Statistical Estimation
2.2.1
Point Estimation
2.2.2
Confidence set
2.3
Constructing estimators
2.3.1
Method of Moments
2.3.2
Maximum Likelihood Estimation
2.4
Empirical Distribution
2.5
Statistical Functionals
2.6
Bootstrap
3
Parametric Inference (Parameter Estimation)
3.1
Method of Moments
3.2
Method of Maximum Likelihood
3.3
Bayesian Approach
3.4
Expectation-Maximization Algorithm
3.5
Unbiased Estimators
3.6
Efficiency: Cramer-Rao Inequality
3.7
Sufficiency and Unbiasedness: Rao-Blackwell Theorem
4
Hypothesis Testing
4.1
Neyman-Pearson Lemma
4.2
Wald Test
4.3
Likelihood Ratio Test
4.4
Comparing samples
PART 3: Models
5
Linear Least Squares
5.1
Simple Linear Regression
5.2
Matrix Approach
5.3
Statistical Properties
MATH 310: Mathematical Statistics (brief notes)
2.6
Bootstrap