Mathematical Statistics

Core Topics

Seven core topics in mathematical statistics, systematically building the theoretical foundation of statistical inference

Beginner

Mathematical Statistics Fundamentals

Master the foundations of mathematical statistics: populations, samples, statistical inference, and core concepts

8 lessons

4-6 hours

Key Content:

Subject positioning & core logic
Population vs sample concepts
Empirical distribution functions
Statistics construction principles
Independence & unbiased estimation
Application areas & real-world examples

Intermediate

Common Distribution Families & Properties

Master probability distribution families, exponential family theory, and their applications in statistical inference

12 lessons

6-8 hours

Key Content:

Basic distributions: binomial, Poisson, normal, exponential
Advanced distributions: gamma, chi-square, t, F, beta
Exponential family theory and properties
Distribution relationships and transformations
Real-world applications and statistical modeling
Parameter estimation and inference methods

Advanced

Point Estimation & Cramér-Rao Inequality

Master statistical estimation theory, evaluation criteria, construction methods, and efficiency bounds

10 lessons

8-10 hours

Key Content:

Point estimation fundamentals and evaluation criteria
Method of moments, MLE, and least squares estimation
Unbiasedness, efficiency, consistency, and MSE
Fisher information and Cramér-Rao lower bounds
UMVUE theory: Rao-Blackwell and Lehmann-Scheffé
Asymptotic properties and efficiency analysis

Advanced

Sufficient & Complete Statistics

Explore sufficient statistics, complete statistics, and their applications in optimal statistical inference

8 lessons

6-8 hours

Key Content:

Sufficient statistics: definition and intuition
Fisher-Neyman factorization theorem and applications
Complete statistics and uniqueness properties
Rao-Blackwell theorem for variance improvement
Lehmann-Scheffé theorem and UMVUE construction
Basu's theorem and independence properties

New!

Intermediate

Confidence Intervals & Interval Estimation

Master interval estimation theory, confidence intervals construction, and pivotal quantity methods for statistical inference

10 lessons

6-8 hours

Key Content:

Interval estimation fundamentals and coverage probability
Confidence levels, confidence coefficients, and interpretation
Pivotal quantity method for interval construction
Normal population confidence intervals (mean and variance)
Two-sample confidence intervals and hypothesis testing connections
Large sample approximations and Bootstrap methods

New!

Intermediate

Hypothesis Testing & Statistical Inference

Master statistical hypothesis testing: error analysis, test construction, common tests, and generalized likelihood ratio methods

12 lessons

8-10 hours

Key Content:

Null and alternative hypotheses construction principles
Type I and Type II errors, power function, Neyman-Pearson principle
U-test, t-test, chi-square test for normal populations
Two-sample tests and F-test for variance comparison
Generalized likelihood ratio test (GLRT) and asymptotic theory
Confidence intervals and hypothesis testing duality

New!

Intermediate

Nonparametric Hypothesis Testing

Master distribution-free statistical tests: sign tests, rank-based methods, goodness-of-fit tests, and independence analysis

14 lessons

9-11 hours

Key Content:

Distribution-free testing principles and robustness advantages
Sign test for median testing and paired sample comparisons
Wilcoxon rank sum test for two independent sample comparison
Wilcoxon signed rank test for paired samples with rank information
Chi-square goodness-of-fit tests and Kolmogorov-Smirnov tests
Chi-square independence tests and run tests for randomness

New!

Intermediate to Advanced

Bayesian Statistics & Inference

Master Bayesian statistical inference: prior distributions, posterior analysis, credible intervals, and prediction theory

12 lessons

8-12 hours

Key Content:

Bayesian inference fundamentals and parameter randomness
Prior distribution construction: moments, quantiles, and expert methods
Conjugate priors and posterior distributions (Beta-Binomial, Gamma-Poisson)
Bayesian point estimation: mean, median, and mode estimators
Credible intervals and posterior uncertainty quantification
Bayesian prediction and decision-theoretic applications
Empirical Bayes and hierarchical modeling approaches
MCMC methods and computational Bayesian inference

Core Topics

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Learning Path

Learning Sequence:

Learning Features

Theory & Practice Integration

Progressive Learning

Master Core Methods

Start Your Mathematical Statistics Learning Journey