Syllabus summary

Course identity

Mathematical Statistics I (MATH 75063) — 3 credits, two 75-minute graduate meetings per week.

Purpose

Develop the theory that explains why statistical procedures work, when they fail, and how better ones are built: formulate a model and an estimand, derive the distribution of a statistic, construct estimators by moments, likelihood, and posterior reasoning, and say honestly what the resulting uncertainty does and does not cover.

Outcomes overview

  • Formulate a statistical problem in terms of data, a sample space, a probability model, a parameter, an estimand, and explicit assumptions.
  • Distinguish a parameter of the observed-data distribution from a scientific or causal target that may require additional identification assumptions.
  • Derive marginal, conditional, and transformed distributions, including distributions of statistics and order statistics.
  • Use exact sampling distributions, pivots, convergence results, the central limit theorem, and the delta method to characterize uncertainty.
  • Construct estimators using the method of moments, maximum likelihood, and Bayesian posterior summaries.
  • Evaluate estimators using bias, variance, mean squared error, consistency, efficiency, loss, and risk.
  • Compute and interpret the score, observed and expected Fisher information, and likelihood-based approximations.
  • Apply sufficiency, the factorization theorem, Rao-Blackwell improvement, completeness, and information bounds.
  • Derive posterior and posterior-predictive distributions for foundational conjugate models and compare them with frequentist procedures.
  • Use R to simulate sampling distributions, optimize likelihoods, examine asymptotic behavior, and evaluate finite-sample performance.
  • Audit generated mathematical arguments and code by identifying assumptions, testing special cases, reproducing calculations, and correcting errors.
  • Communicate a rigorous statistical argument in clear mathematical prose, with appropriate notation, assumptions, and limits on the conclusion.

Materials and access

All course materials are provided on this site at no cost. Open sources in support:

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.