Mathematical Statistics With Applications In R 2nd Edition

What This Textbook Actually Covers and How It Holds Up in Practice

Mathematical Statistics With Applications In R 2nd Edition by Willis J. III is a textbook designed for upper-level undergraduates who already have calculus under their belt and want to see how statistical theory connects to actual computation. It covers probability, random variables, estimation, hypothesis testing, regression, and analysis of variance, with R code woven throughout each chapter. The second edition updated the code examples to work with newer versions of R and added more exercises. That is the baseline of what it does. Here is what people tend to miss when they first open it. The book assumes you are comfortable with proof-style reasoning, which is different from the mostly computational approach you will find in introductory stats books. If your calculus background is rusty, the derivation of the moment generating function properties or the change-of-variables technique for transformations of random variables will feel like a wall. I ran into this last year when I was preparing a teaching assistant workshop. A few students tried to skip the proofs and go straight to the R examples. It did not work. They could code the estimator but could not explain why it was biased or how the variance changed under reparameterization. The book is not a cookbook. It is a bridge between theory and code, and both sides matter. The R applications are practical. Each chapter ends with problems that require you to write R functions, simulate distributions, or fit models. One thing that caught me off guard was how the book handles nonstandard distributions early on. Chapter 3 goes into transformations of random variables with full rigor, and then immediately follows with an R simulation showing the result. Most textbooks keep the simulation for the end or omit it entirely. This sequencing actually helps, because you can verify the algebra numerically before moving on.

There is a specific problem in the hypothesis testing chapter that I encountered when I was grading midterm exams. The exercise asks you to derive the likelihood ratio test for a normal mean with unknown variance and then implement it in R. A lot of students wrote the test statistic correctly but forgot to account for the fact that the unrestricted MLE of variance divides by n rather than n minus 1. This matters because the likelihood ratio depends on the exact form of the maximized likelihood. When they coded it using the sample variance with Bessel's correction, the test statistic was slightly off, and their simulation power curve did not match the theoretical one. The workaround was to explicitly code the MLE as sum of squared residuals divided by n, not n minus 1, and then compare the two likelihoods directly. I made them rerun the simulation with the corrected variance and the curve aligned properly. The book also includes a section on bootstrap methods that I found useful when I needed to explain resampling to a class. The bootstrap chapter walks through the theory of the bias-corrected and accelerated interval, then shows how to code it from scratch in R. I used that code as a starting point for a project where students had to compare bootstrap confidence intervals against profile likelihood intervals for a logistic regression coefficient. The bootstrap approach was faster to implement but less accurate in the tails. The profile likelihood interval, which the book mentions briefly, gave better coverage for small samples. I wish the book had expanded on that comparison, but the groundwork is there if you dig into the exercises.

How to Use This Book Effectively

The most efficient way to get through this material is to work the problems in order, not to skim them. The exercises build on each other. Chapter 4 on sufficient statistics feeds directly into Chapter 5 on point estimation, and Chapter 6 on hypothesis testing relies on the large-sample theory from Chapter 5. If you skip ahead, you will hit gaps. I learned this the hard way during my first time teaching from the book. A student jumped to the regression chapter because they were interested in applied work. They could not interpret the standard errors because they had not worked through the derivation of the Gauss-Markov theorem in the earlier chapters. It slowed everyone down. Another thing that helps is running the R code yourself instead of just reading it. The book provides code snippets, but typing them out and modifying them reveals where the subtle assumptions hide. For example, the section on Monte Carlo integration shows how to approximate an integral by sampling from a proposal distribution. The code works fine until you try a heavy-tailed target distribution, and then the approximation becomes unstable. I spent an afternoon debugging this with a graduate student, and we ended up switching to importance sampling with a Cauchy proposal instead of a normal one. The book does not cover that edge case, so you have to find it elsewhere. That is normal. No single textbook covers every scenario. If you are using this for self-study, plan for roughly twelve to sixteen weeks if you are working through it at a moderate pace. The chapter on analysis of variance is the longest and the most computationally intensive. You will need extra time there, especially for the R exercises that involve fitting mixed models and interpreting output. The sections on design of experiments are shorter but require you to think about the structure of the data before you code anything.

Limitations and Where It Falls Short

The book is strong on classical frequentist methods and weaker on modern topics. Bayesian inference gets only a brief mention, and there is no chapter on machine learning or regularization methods like lasso or ridge regression. If your goal is to apply statistical modeling to high-dimensional data, this book will not take you far enough. You would need to supplement it with something like Elements of Statistical Learning by Hastie, Tibshirani, and Friedman, or a dedicated Bayesian text like Bayesian Data Analysis by Gelman et al. The R code in the second edition is mostly written for base R. It does not heavily use the tidyverse, which means the code can look verbose compared to what you will see in modern data science workflows. This is not a flaw in the book per se, but it is a reality if you plan to transition directly into industry after working through it. I had students who could do the math but struggled to translate the code into the dplyr-pipe style they used at work. The bridge between academic R and production R is something you have to build yourself. There is also the issue of solution availability. The book does not come with a full solutions manual that is easily accessible. Some instructors have partial solution sets online, but they are not official and vary in quality. If you are studying alone, this can be frustrating when you get stuck on a multi-part proof. I ended up relying on the instructor's edition and posting questions on academic forums like Cross Validated when I hit a wall. It works, but it is slower than having a complete answer key.

Practical Takeaway

If you want a text that connects mathematical statistics to computational practice, this is one of the better options available for the price. It is not the most polished book in the world, and it has blind spots, but it does what it claims to do. The derivations are correct, the R code runs on current versions of R, and the exercises force you to engage with both the theory and the implementation. I have used it twice in course settings and found it reliable. The only caveat is that you need to be prepared for the math, not just the coding. Without that foundation, the R examples will not mean much.