Syllabus summary

Course identity

Mathematical Statistics II (MATH 75163) — 3 credits, three contact hours a week of graduate lecture and workshop.

Purpose

Treat a statistical procedure as a decision rule with assumptions and operating characteristics rather than a mechanical route to a p-value: derive how a procedure is built, calibrate what it actually does, stress it where its assumptions fail, and compare exact, asymptotic, resampling, robust, and Bayesian answers to the same question.

Outcomes overview

  • Formulate statistical inference as a decision problem with an estimand, action space, loss function, procedure, and risk.
  • Define and calculate size, level, power, type I error, type II error, and power functions, and explain what each quantity does and does not establish.
  • Apply the Neyman-Pearson lemma and monotone likelihood-ratio arguments to construct most powerful and uniformly most powerful tests.
  • Use the duality between tests and confidence sets to derive and interpret exact and approximate procedures.
  • Derive and compare likelihood-ratio, score, and Wald tests, including their regularity assumptions and finite-sample limitations.
  • Handle nuisance parameters using conditioning, profiling, invariance, or asymptotic approximation when appropriate.
  • Derive central results for normal linear models using projections and quadratic forms, including t, F, regression, and analysis-of-variance procedures.
  • Explain and evaluate family-wise error control, false-discovery-rate control, multiple comparisons, and introductory selective-inference concerns.
  • Construct and justify randomization, permutation, and exact conditional procedures from the assignment or sampling mechanism.
  • Use bootstrap and jackknife methods, and assess their calibration through theory and simulation.
  • Apply foundational nonparametric rank procedures and compare their assumptions, targets, and efficiency with parametric alternatives.
  • Evaluate robustness using influence, contamination, breakdown, M-estimation, and sandwich-style variance ideas.
  • Derive and compare Bayes rules, minimax rules, admissible procedures, shrinkage estimators, and predictive decisions in foundational settings.
  • Distinguish evidence against a null model, posterior model probability, predictive performance, and scientific importance.
  • Use R to estimate operating characteristics, compare procedures, diagnose approximation failure, and reproduce inferential results.
  • Audit AI-generated mathematical and computational work by checking assumptions, theorem conditions, edge cases, code execution, and independent derivations.
  • Communicate a defensible inferential conclusion with explicit assumptions, calibration claims, sensitivity results, and limitations.

Materials and access

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Logistics

The public materials for this course are in development. Dates, policies, assessments, and readings on this site are provisional and may change, and the term shown is a planning assumption rather than a scheduled section. The learning management system remains authoritative for section logistics and graded details.