Topics in Mathematical Statistics
1
Introduction
2
Measure-based Probability theory
2.1
Probability Spaces and Random Elements
2.1.1
Recap: Measure theory
2.1.2
Measurable Functions
2.2
Integration and Differentiation
2.2.1
Lebesgue integral
2.2.2
Radon-Nikodym Derivative
2.3
Conditional Expectation
2.3.1
Conditional Expectation
2.3.2
Independence
2.3.3
Conditional Distribution
2.3.4
Markov chain and Martingale
2.4
Asymptotical Theory
3
Statistical Decision Theory
3.1
General Framework
3.2
Point Estimators
3.2.1
UMVUE
3.2.2
U-Statistics
3.2.3
V-Statistics
3.3
Hypothesis Tests
3.3.1
UMP Tests
3.3.2
Tests in Parametric Models
3.3.3
Tests in Nonparametric Models
3.4
Confidence Sets
4
Estimation in Parametric Models
4.1
Bayes Estimators
4.1.1
Markov chain Monte Carlo
4.1.2
Asymptotic efficiency of Bayes Estimators
4.2
Method of Maximum Likelihood
4.2.1
MLE
4.2.2
Quasi-likelihoods and conditional likelihoods
4.2.3
Asymptotic efficiency
5
Estimation in Nonparametric Models
5.1
Distributional Estimators
5.2
Statistical Functionals
5.3
Sample quantiles
5.4
Generalized Estimating Equations
5.4.1
Framework
5.4.2
Consistency
5.4.3
Asymptotic normality
5.5
Generalized Methods of Moments
6
Final Words
References
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Topics in Mathematical Statistics
4.2
Method of Maximum Likelihood
4.2.1
MLE
4.2.2
Quasi-likelihoods and conditional likelihoods
4.2.3
Asymptotic efficiency