Statistics for Mathematicians: A Rigorous First Course

Statistical reasoning and analysis is applicable wherever there is data, and today this seems to be everywhere. Despite the importance of the field, it seems that statistics often does not have the same initial appeal to many students attracted to mathematics as do, say, number theory, combinatorics or dynamical systems. This is interesting, since at the level of current research, probability and statistics do seem to have deep relations with each of the fields of number theory, combinatorics and dynamical systems. Perhaps this is because many books on elementary probability and statistics, notwithstanding Feller’s beautiful Introduction to Probability Theory and its Applications, seem to be written in quite a different style from texts on other major areas of mathematics.

In Statistics for Mathematicians, Victor M. Panaretos addresses this issue and works hard to make statistics appealing to all mathematicians. First off, this book is beautifully written and very well done. Definitions and theorems stand out clearly and there are interesting exercises and very nicely produced figures. I think that mathematics students, particularly those of an analytical bent, will enjoy this book. The main focus of the text is on a rigorous presentation of one-parameter inference. A background in basic probability is assumed, although there is a pretty nice summary of the most important and relevant results from probability in an appendix. Namely, the reader should know the meaning of terms such as random variable and distribution at the level of a first course.

In the opposite direction, Statistics for Mathematicians does not contain much at all on the practical analysis of data or the use of computation in statistical modeling. Thus, this book may not serve well for some practically minded statisticians. Nonetheless, I do think that this book is worth looking at to get a sense of some of the intrinsic mathematical beauty of statistics. The author admirably achieves his goal “to provide an introduction to the subject tailored to the mindset and tastes of Mathematics students, who are sometimes turned off by the informal nature of Statistics courses.”

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.