Welcome to MATH 563 Mathematical Statistics

Spring 2026

Course Description

Theory of sampling distributions; principles of data reduction; interval and point estimation, sufficient statistics, order statistics, hypothesis testing, correlation and linear regression; introduction to linear models.

Instructor: Fred J. Hickernell

Fred J. Hickernell

Brief bio: Fred J. Hickernell is professor of applied mathematics. His research focuses on improving the efficiency of computer simulations and developing justifiable stopping criteria. He has spent most of his career at Hong Kong Baptist University and Illinois Institute of Technology, often in academic administrative roles. He has served on the editorial boards of the Journal of Complexity, Mathematics of Computation, and the SIAM Journal on Numerical Analysis. He is a Fellow of the Institute of Mathematical Statistics and the recipient of the 2016 Joseph F. Traub Prize for Achievement in Information-Based Complexity.

Hickernell speaks Cantonese and enjoys Chinese food. He is married with adult children. He enjoys discussing matters of substance. His most important identity is a disciple of Jesus.
Teaching Assistant: Nishi Thakur

Textbook: G. Casella & R. L. Berger, Statistical Inference, 2nd ed. (2002)

Recommended resources
Prerequisites
  • Multivariate calculus
  • Linear algebra
  • Calculus-based probability
  • Comfortable wiht proofs
  • Comfortable with code
Objectives

You will learn

  • The concept of statistical inference and the difference between population characteristics and sample estimates;
  • The probabilistic basis for statistical inference and the qualities of a good estimator;
  • How to correctly perform hypothesis tests and construct confidence intervals;
  • The appropriateness of asymptotic considerations;
  • Use statistical software—primarily Python or R—to perform basic computations for statistical inference and sampling; and
  • How to communicate the results of their statistical analyses of substantial data sets through explanatory text, tables and graphs.
Where to find it
This course website Canvas
Big Questions Grades
Schedule Assignments and Submissions
Jupyter Notebooks Announcements
Homework Discussions
Tests and Exams
Policies

Course Outline

Introduction
  • What is statistical inference?
  • Review of probability
    • Random variables and their distributions
    • Populations and samples
    • Sample statistics
Properties of a random sample
  • Sampling from Normal distribution
  • Order statistics
  • Convergence concepts
Principles of data reduction
  • The sufficiency principle
  • The likelihood principle
Point estimation
  • Methods of finding estimators: moment, MLE, Bayes
  • Methods of evaluating estimators: MSE, bias, sufficiency, Rao-­Blackwell theorem, loss function optimality
Hypothesis testing
  • Methods of finding tests: likelihood ratio test, Bayesian test
  • Methods of evaluating tests: error probabilities, power function, p-alues, Neyman-­Pearson Lemma
Interval estimation
  • Pivoting method
  • Size and coverage probability
Asymptotic evaluation
  • Point estimation: consistency, efficiency, comparisons
  • Asymptotic distribution of LRTs / confidence intervals
Introduction to linear models
  • Simple linear regression: least squares
  • One-­way ANOVA

Assessment

  • Homework: 15% (will drop the lowest 2 homework scores)

  • Midterm Tests: 25% (2/3 times best score + 1/3 times second best score)

  • Project: 20% (see project page for details)

  • Final Exam: 40%

  • Extra Credit: up to 10%
    Beginning January 20, if you are the first to find an error in

    • My lecture slides, Jupyter Notebooks, or assignments at least 24 hours after I have presented them in class, or
    • The textbook,

    Then I will award you 0.5 or 1 extra credit points, up to a maximum of 10 points.  

    For course materials hosted on GitHub, you may submit a pull request correcting the error. Alternatively, you may report the error by email with sufficient detail. Extra credit points are added to your final weighted total. Note that the weighted total may not correctly reflect the extra credit until grades are recorded for all categories.