Probability and Mathematical Statistics: Theory, Applications, and Practice in R

As the title implies, this is intended for a two-semester upper level undergraduate sequence in Probability and Mathematical Statistics. All of the standard canon and a fair bit more is covered carefully, with many good examples and exercises. There is a chapter on catagorical data tests, and one on analysis of variance. There is no discussion of regression. Meyer includes frequent R code snippets for simulations, checking probability answers, demonstrations, and bootstrap confidence intervals. There is a remarkable emphasis on hypothesis testing and power calculations throughout the book. 

 

The structure of the text is one of its distinguishing characteristics. It is broken up into 60 quite short chapters - anywhere from four to sixteen pages, most between six and twelve - each with an easily digestible topic (from an undergraduate’s point of view). Most chapters would serve as the basis of a typical 50 or perhaps 75 minute lecture. Virtually all major distributions are given their own chapter. Of course, you may assign them as reading or omit some as you see fit. Chapters end with a colored box collecting in one place the results and new information in that chapter. There are a remarkable number of nice exercises in every chapter. I like that Meyer chooses not to bury new definitions or terminology among them.  The writing style is quite clean and to the point, without being too brief. I have taught from easily a dozen different such texts over the years, and I can say the exposition of the mathematics is as clean and clear as I have seen. There is little clutter to the exposition: no figure or equation numbers, highlighted text, boxes, sidebars, or all the other miscellaneous baggage publishers seem to think is necessary in a modern text. Proofs are broken down whenever possible into the simplest case, then extended to the more general conditions. The exposition is helped by the many figures in multiple colors. Despite the lack of figure and equation numbers, I was never at a loss for which figure was being referred to in the text; the placement was carefully chosen.

 

The order of topics and emphases are a bit nonstandard. There is a strong emphasis from early on (Chapter 6) on hypothesis testing and power calculations (pet peeve: there are many usages of ‘accept H 0 ’). Nearly every hypothesis test example or exercise includes power calculations as well. Such exercises are larded throughout the text. Frequent R code snippets are used to check calculations or replace them in tedious situations. Confidence intervals, on the other hand, do not appear until much later (Chapter 48). There is a good discussion with simulations of actual coverage for populations that are not normal. Many of the various techniques are compared using R simulations for various population distributions.Bayesian analysis and credible intervals are also covered.

 

Meyer provides no real introduction to the instructor, or to the student. There is no guidance on recommended sequences or dependencies among chapters, nor on the level of R knowledge needed (very little is required). An instructor should not have much of an issue with identifying core and ancillary chapters and dependencies among them, but some guidance should have been included. There is little commentary, leaving much of the nuances of application or motivation to the instructor. In the body of the text, there are scant motivations for many of the results. The Bayes estimation chapter was dense and would be quite confusing for students; I would omit that chapter and the one on credible intervals (which don’t have to be intervals, contrary to Meyer). There is little discussion of (lack of) robustness of estimators or tests. For example, no discussion of problems with tests for variance. There is no critique at all of p-values, or real rationale for why Bayesian approaches might be of interest.

 

That said, all in all I like the text for its clean and clear writing, as well as short and compartmentalized approach to the material. The exercises are extensive and well thought out. Any instructor will pick and choose material; the short chapters make this quite easy to do. The lack of more motivational commentary allows, as a colleague once commented to me, for the instructor to be ‘the good guy’ and provide that information themselves to students. Meyer's text is definitely worth consideration for your courses.

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James Helmreich teaches at Marist College in Poughkeepsie NY and works in statistics, graphics, and pedagogy. His email address is james.helmreich@marist.edu, and his website is https://foxweb.marist.edu/users/james.helmreich/.