There are a variety of textbooks on nonparametric statistics. In the early days, these techniques were defined negatively by their failure to make many assumptions about the population from which a sample was drawn. Then it was realized that many of these techniques amounted to applying parametric techniques to the ranks of the data. Ignoring ties for the moment, the population ranks follow a uniform distribution, in which case we need no assumptions about its shape — we know what it is. The book at hand takes this approach but goes farther and also contrasts parametric and nonparametric statistics in terms of the L_{2} and L_{1} norms. That is not a translation into MAAspeak but exactly how the book describes things. So, we might consider this book to be “math. stats. for nonparametrics”. In that category, it hasn’t much competition.
MAA members can be expected to be familiar with the linear algebra and matrix notation in this book, as well as the theoremproof presentation and the norms involved. A good, modern regression course, with an emphasis on diagnostics and the use of matrices, would be one statistical prerequisite, as the presentation often parallels least squares with alternatives — very helpful for those who already know the least squares side of the story. Familiarity with the common nonparametric techniques would allow one to benefit from the many times they are referenced but not explained in detail. Finally, a mathematical statistics course might contribute knowledge of the sorts of properties statisticians consider desirable in estimators. The theorems proven are not always the ones a mathematician might think worthy of attention, but are driven instead by applications and history.
The main audience for this text is probably statistics departments with Ph.D. programs. MAA members who might be interested include those who are called upon to teach a nonparametric statistics course and who want to understand what is really going on. It carries a significant list of prerequisites, but it does really use and integrate them, making this book a fine capstone course in nonparametric statistics.
After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from fulltime classroom work. He now teaches statistics online at statistics.com and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.
