Learn Twenty-two lessons deriving estimators, tests, intervals and regression from probability models. Exact finite-sample results, regular asymptotic approximations and model-dependent decisions are distinguished. Mixed assessment, independent analysis and proof review remain separate requirements; topic coverage alone does not establish semester equivalence.
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Estimator risk begins with a sampling distribution under stated assumptions.
The expectation is p and the variance is p(1−p)/n. Independence removes covariance terms; identical success probabilities make the common parameter p meaningful.
Parameter, statistic and random interval have different roles in mathematical inference.
The data and resulting interval vary under repeated sampling; the target parameter is fixed. Coverage is a property of the procedure under a specified model, not a probability assigned to that fixed parameter after observing one interval.
Conditional expectation underlies Rao–Blackwell improvement and several statistical decompositions.
The conditional mean is the orthogonal squared-error predictor among functions of the conditioning information. Expanding the square yields an irreducible conditional-variance term plus the squared difference from that mean.
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Mathematical statistics: derivation, decisions and model checks
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Statistical investigation: vehicle efficiency, model revision and changed targets
Sustained investigation with separate transfer evidence, saved revisions and human review. It does not change lesson participation requirements.
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Separate the sample space, parameter, statistic and sampling design.
6 practice tasks.
Use likelihood ratios and distinguish observationally equivalent parameters.
6 practice tasks.
Derive MSE and distinguish pointwise risk from universal preference.
6 practice tasks.
Use convergence arguments and identify weak points in moment matching.
6 practice tasks.
Handle support constraints, boundary maxima and nonexistence.
6 practice tasks.
Use conditional distributions and dominated factorization.
6 practice tasks.
Derive moments from the log normalizer and respect the natural parameter space.
6 practice tasks.
Derive risk reduction and show where sufficiency enters.
6 practice tasks.
Prove a complete-family example and the Lehmann–Scheffé conclusion.
6 practice tasks.
Derive an information bound and inspect regularity failures.
6 practice tasks.
Use score expansion and distinguish a theorem from a diagnostic.
6 practice tasks.
Use first- and second-order expansions where appropriate.
6 practice tasks.
Use orthogonal decomposition to prove t and chi-square pivots.
6 practice tasks.
Prove the likelihood-ratio ordering and handle randomization.
6 practice tasks.
Use monotone ratios and stochastic ordering to control a composite null.
6 practice tasks.
Derive a likelihood-ratio statistic and mark the limits of Wilks calibration.
6 practice tasks.
Connect acceptance sets, confidence sets and prospective power.
6 practice tasks.
Derive conjugate updating and separate prior-sensitive decisions from sampling guarantees.
6 practice tasks.
Whiten normal vectors and identify independent orthogonal quadratic forms.
6 practice tasks.
Separate fixed-design algebra from normal-error inference.
6 practice tasks.
Use leverage and residual covariance to diagnose model departures.
6 practice tasks.
Derive empirical resampling and diagnose nonsmooth or dependent failures.
6 practice tasks.