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.

To build the spiral structure, I have rather radically revised the order in which I cover topics, relative to the order one usually sees in a classic math-stat textbook. That textbook might have one chapter that illustrates the properties of discrete probability distributions using, e.g., the binomial, geometric, and Poisson distributions, then a second chapter that illustrates the properties of continuous distributions such as, e.g., the normal and gamma distributions, with neither chapter showing how these distributions are applied in statistical inference. Here, major distributions are broken out into their own chapters, and within each I revisit fundamental concepts: probability mass and density functions, cumulative distribution functions, statistics, sampling distibutions, point estimation, interval estimation, and hypothesis testing, etc. Then, as I move from chapter to chapter, I cover concepts at greater depth. As a concrete example: when discussing point estimation in Chapter 1, I introduce the concepts of bias, variance, and using the likelihood function to define estimators; when I return to point estimation in Chapter 2, I review these concepts (and derive the MLEs for normal distribution parameters), then add the concepts of consistency, Fisher information, the Cramer-Rao lower bound, and the asymptotic distribution of maximum likelihood estimates. Then in Chapter 3, I add in sufficient statistics and likelihood factorization, along with the minimum variance unbiased estimator. Etc.

Another important difference between this textbook and older, more established math-stat textbooks is that I utilize R for coding visualizations, analyses, and simulations. It is expected that this will help students understand concepts more readily; it also allows me to broaden the “problem space” beyond typically used, analytically tractable distributions. (But I note that the newest generation of textbooks often employ enhanced computation…so it is really the spiral structure that makes this textbook fundamentally different.)

If you are an instructor, feel free to utilize aspects of this book for your own class(es), but please do not share this document without my expressed consent. If you have comments or questions (or you want an updated version), please send email to the address given below.

Peter Freeman

pfreeman@cmu.edu

July 2024