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
2.1
Simple Random Sample
2.2
Standard Random Sample
3
Parametric Inference (Parameter Estimation)
3.1
Point Estimation
3.2
Confidence set
3.3
Method of Moments
3.4
Maximum Likelihood Estimation
3.4.1
Consistency
3.4.2
Asymptotic normality
3.4.3
Efficiency
3.5
(Optional) Expectation-Maximization Algorithm
3.6
Bayesian Approach
3.7
Comparing Estimator / Decision Theory
4
Hypothesis Testing
4.1
Procedure
4.2
Neyman-Pearson Lemma
4.2.1
Randomized Test (Optional)
4.3
Uniformly Most Powerful Test
4.4
Wald Test
4.5
Likelihood Ratio Test
4.6
Comparing samples
MATH 310: Mathematical Statistics (brief notes)
4.5
Likelihood Ratio Test