Mathematics / core
Mathematical statistical inference
Derive estimators, tests, and intervals from probability models while separating assumptions from guarantees.
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Module 01 core
Mathematical statistical inference
Derive estimators, tests, and intervals from probability models while separating assumptions from guarantees.
- 01
Statistical models and identifiability
Separate observable distributions, unknown parameters, and quantities the experiment can actually identify.
25 min - 02
Likelihood and sufficient statistics
Use factorization and conditional laws to identify what a statistic preserves about a parameter.
26 min - 03
Exponential families and completeness
Connect natural parameters, log normalizers, sufficient statistics, and a uniqueness property of unbiased estimation.
28 min - 04
Unbiased estimation and Rao–Blackwell improvement
Average away irrelevant randomness and prove the resulting variance improvement with conditional expectation.
28 min - 05
Fisher information and Cramér–Rao bounds
Derive an exact variance lower bound from a score covariance and learn when its assumptions fail.
30 min - 06
Maximum likelihood and score equations
Find global likelihood maximizers, handle boundaries, and derive a regular asymptotic normal approximation without promising exact finite-sample behavior.
29 min - 07
Neyman–Pearson testing and power
Build a most-powerful simple-hypothesis test and calculate exact size, randomization, and power.
30 min - 08
Confidence sets and test inversion
Construct confidence procedures by retaining nonrejected parameter values and interpret coverage as a repeated-experiment guarantee.
29 min - 09
Likelihood-ratio tests and pivots
Separate a likelihood-ratio construction from its calibration and distinguish exact pivots from Wilks approximations.
30 min - 10
Decision theory and minimax risk
Compare full risk functions and prove a finite-sample minimax result using a constant-risk Bayes estimator.
31 min