The following quote from my ICOTS 2026 conference proceedings paper The Future of Statistical Inference in the Classroom: Aligning Theory With Computation sets the scene well:

“[T]he term ‘mathematical statistics’ denotes a calculus-based course sequence that covers concepts of probability and statistical inference, generally over the course of an academic year. This course sequence is typically targeted to those pursuing a degree in statistics and/or data science. The concepts covered within it—elementary probability theory, random variables and univariate and multivariate probability distributions, statistics and sampling distributions, point and interval estimation, hypothesis testing, and statistical modeling—are largely if not entirely the same now as those that would have been covered in a mid-20th century instantiation of the sequence. Panelists at the JSM session Is the Math Stat Course Obsolete?…referred to these concepts collectively as the ‘Hogg and Craig syllabus’ (while specifically referencing the 1965 second edition of Hogg and Craig’s Introduction to Mathematical Statistics). In our opinion, mathematical statistics is not obsolete: knowledge of probability distributions and how they are used to make inferences about populations is fundamental to the enterprise of statistics! However, aspects of the Hogg and Craig syllabus that still permeate mass-market textbooks, such as those that rely on the use of large-sample approximations and statistical tables, have become obsolete and need to change.”

Textbook

Modern Probability and Statistical Inference (Illustrated with R)

From the preface:

“I have developed this textbook for a new mathematical statistics course sequence for statistics students at Carnegie Mellon University (numbered 36-235 and 36-236), first piloted in Fall 2022 and Spring 2023. The primary difference between it and others that are commonly used in math-stat classes is that I explicitly implement a spiral-learning framework, with concepts that are usually covered in relative isolation elsewhere being repeatedly covered here (such as, e.g., point estimation).

“My use of a spiral-learning framework is motivated by the observation that many students fixate on mathematics and coding and pay less attention to statistical concepts, with the details of concepts that are seen once often quickly forgotten. The spiral approach is meant to result in enhanced conceptual retention.”

Manuscript

Modernizing How Students Compute Interval Estimates in Mathematical Statistics Courses

Abstract:

The calculation of confidence intervals is a fundamental aspect of statistical inference that undergraduate students are expected to master when taking calculus-based mathematical statistics courses. In this work, we describe an interval estimation framework that we present to statistics majors at Carnegie Mellon University, one that supersedes the traditionally taught pivotal method. When utilizing this framework, which we dub “cdf inversion,” students determine the cumulative distribution functions of interval-estimating statistics through any of a number of approaches, including analytic derivations, numerical simulations, and the bootstrap; importantly, these statistics need not be pivotal quantities. Given the cdf, students then derive interval bounds analytically (when possible) or by coding a numerical root-finding algorithm. We find that utilizing cdf inversion offers numerous advantages: for instance, by removing the need for pivotal quantities, we greatly expand the space of interval-estimation problems that mathematical statistics students can solve, while at the same time making solutions easier to formulate. In addition, because it is possible for students to fully implement cdf inversion via code, we feel that elements of the cdf-inversion framework can, with proper scaffolding, be incorporated into introductory statistics and data science class settings.

Additional Materials

One would find on my GitHub site the following repositories with material related to mathematical statistics: