This book is about the probability theory that is useful for statistics, done without measure theory. Its prerequisites are a decent background in analysis (the author suggests baby Rudin), and enough background in statistics to make the topics meaningful. It covers all the standard topics you would expect: distributions and their properties, common families of distributions, expectation (conditional and otherwise), normal distribution theory, limit theorems, etc. It also contains a few topics that are not so standard for this type of book: exchangeability, martingales, stochastic processes. In my opinion, the highlights of the book are chapters 9, 10, 13 & 14, on Approximation of Integrals, Orthogonal Polynomials, Approximations to the Distributions of More General Statistics, and Higher Order Asymptotic Approximations. These chapters are very interesting, well written, and great introductions to their subjects. In fact, due to chapter 10’s references, I stopped at the library to take out Andrews, Askey, and Roy’s wonderful Special Functions.
Each chapter has suggestions for further reading and many (20-30) exercises of varying difficulty. An instructor will have no trouble choosing problems for assignments. Appendices cover some basic mathematical facts, and set some notation.
The book could easily be used as a text or as a supplement to, for example, Casella and Berger’s Statistical Inference. I can see many students using it to learn the details for topics that the standard texts don’t have space to include.
The material is well written, proofs are easy to follow, and motivation is clear. There are few typos that jumped out at me, but two that did are as follows. Theorem 1.5 states that the set of discontinuity points of a distribution function is countable, but the proof seems muddled to me. Better to just state (or perhaps prove in an appendix) that a countable union of countable sets is countable. Then, in Theorem 9.17, the definition of P(x) has a missing closing bracket for the floor function, which is then carried throughout the rest of the section.
Sample pages can be seen in Google Books.
Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. He is currently writing about the theory of Mallows permutations.
1. Properties of probability distributions
2. Conditional distributions and expectation
3. Characteristic functions
4. Moments and cumulants
5. Parametric families of distributions
6. Stochastic processes
7. Distribution theory for functions of random variables
8. Normal distribution theory
9. Approximation of integrals
10. Orthogonal polynomials
11. Approximation of probability distributions
12. Central limit theorems
13. Approximation to the distributions of more general statistics
14. Higher-order asymptotic approximations.