This course introduces students to the analytical tools needed to establish the properties of common statistical methods. Why do sampling distributions of many statistics become bell-shaped with large sample sizes? When is the sample mean the best estimator for the population mean? What do we mean by "best" estimator?

This course takes a likelihood-based perspective to provide a unified framework to estimation and hypothesis testing. Building a proper foundation for inference, we emphasize the application of theory throughout the course. In additional to classical theory, the course makes use of computing as a bridge between that classical theory and the modern approaches used characterize statistical methods.

The class builds towards preparing students to engage in the statistics literature at an introductory level. This class is particularly suited for those students considering graduate level work in statistics. It does require a firm grasp of probability.

Learning Objectives

As in any statistics course, we emphasize statistical literacy and reasoning. This class also emphasizes the underlying theory of statistical methodology. Specifically, after taking this course, students will be able to accomplish the following tasks:

  1. Describe the role of probability theory when making inference on a population.
  2. Compare and contrast modes of convergence and describe their role in sampling distributions.
  3. Construct a proof to establish a property of an estimator or sequence of random variables.
  4. Given an analysis situation, formulate research questions as measurable statements about parameters in the population.
  5. Associate a given statistical method with various statistical properties such as consistency, unbiasedness, and asymptotic normality.
  6. Clearly communicate statistical theory and its applications through both written and oral presentations.
  7. Appreciate the use of mathematics in statistical theory and compare and contrast mathematical and statistical thinking.
  8. Using fundamental ideas of convergence, likelihood, and probability, research a topic in statistical theory not discussed in the course and educate your peers on the topic.

Course Structure

This course is currently taught as a traditional lecture. The course is divided into 10 modules. Each module has associated assignments. For each module, students can expect three lectures which illustrate foundational concepts in the course. One class period is reserved within each module for a group activity in which students work together to fill out a theoretical result associated with the material. The modules covered include:

  1. Essential Probability for Random Vectors
  2. Expectations
  3. Multivariate Normality and Linear Models
  4. Point Estimation
  5. Sampling Distributions
  6. Inference from a Likelihood Perspective
  7. Introduction to Nonparametric Estimation
  8. Introduction to Bootstrap Theory
  9. Introduction to Bayesian Inference
  10. Capstone Project

The best part of the class is the Capstone Project. Over the course of the term, we cover quite a bit of material regarding the theory underlying many statistical methods. However, there are countless fascinating topics we will leave untouched. The capstone project invites students to explore an additional area in foundational statistics and present a lecture to their peers on the topic. This is often a great celebration of the knowledge gained during the term.

Course Materials

I have not settled on a text for this course and have tried out several. I do supplement the text used with note packets used during lecture and in-class activities.

Statistical computation is a big part of the course; while R is demonstrated and supported, students are free to use any computational software they like (Python, Julia, Matlab).

Some students think of this class as "real analysis for statistics," but that is not quite accurate. However, there is quite a bit of theory and proofs in the class. The best resource in a class like this are your peers. I will provide additional tips and tools throughout, but it is highly recommended to enter the class thinking about who you might work with throughout the course.