Probability and Computing: Randomized Algorithms and Probabilistic Analysis

Chapter 7 begins with a nice introduction of the importance of asymptotic distribution theory for all statisticians, theoretical and applied. There is excellent motivation and explanation of convergence in distribution (law), probability, and almost surely (with probability one). The central limit theorem (CLT) is adequately covered although in the proof, the o(t) notation is used to reflect terms that contain powers of t that are at least k, but the terms need to possess powers exceeding the necessary k for the result to be true. The CLT is applied to the topic of “within normal limits,” which can be used on a host of examples in health care. Also, the delta method is nicely treated. Chapters 8 and 10 cover point and interval estimation. Examples of early attempts to estimate parameters using data are given including the Greek Hipparchus attempting to estimate variability of the length of a day. There is adequate coverage of method of moments, maximum likelihood, and Bayes estimation with standard examples illustrating each. Comparison of estimators includes bias and mean squared error. The discussion of sufficiency is excellent as is the discussion of uniformly minimum variance unbiased estimators (UMVUEs), giving the appropriate background of Rao–Blackwell, Lehmann– Scheffe, completeness, and the fact that sufficient statistics derived from exponential families are complete. The Cramer–Rao Lower Bound is given along with its usefulness in finding UMVUEs. Robustness is not explicitly mentioned, though M-estimation is covered in a single paragraph. Chapter 10 gives a clear explanation of how to interpret a confidence interval as well as an accurate distinction between classical confidence intervals and Bayesian credible intervals. Confidence intervals are derived for a variety of examples using the pivotal quantity method and a couple of examples illustrate the concept of Bayesian credible intervals. Approximate confidence intervals are explained and illustrated and the bootstrap method is very briefly (1.5 pages) discussed. Chapter 9 gives a thorough development of hypothesis testing starting from the beginnings of the scientific method and the role of discovery in developing hypotheses. The authors present statistical testing as a “proof by contradiction” argument originated by Dr. John Arbuthnot in 1710 and later developed by R. A. Fisher in 1925. The Neyman–Pearson lemma is adequately covered. The Likelihood Ratio Test Principle is never formally stated but is illustrated, though maximum likelihood estimators (MLEs) are occasionally omitted and the algebraic connection between the rejection region based on the derived statistic and that for the test based on the likelihood ratio itself is sometimes unclear. The problem of familywise error is discussed when there is more than one analysis taking place. Nonparametric testing is briefly discussed as well as goodness-of-fit tests and sample size determinations. Nothing on permutation tests is included. Several computational methods are given in Chapter 11 to address situations in which analytical methods at times leave statistical problems unsolved or are too complex to pursue. The authors discuss the Newton–Raphson method to find roots of complicated functions, the expectation maximization algorithm to find MLEs, and several simulation techniques to find estimates of unknown parameters. Unfortunately the positive features of the book are counterbalanced by several serious flaws. Spacing (vertical and horizontal) is not consistent throughout the text and is awkward at times. More compellingly, there are many typos throughout the text and numerous missing periods and commas; a multitude of errors appear in Chapters 1, 3–5, 7, and 9, fewer in the other chapters. (I have provided the authors with an errata list for those mistakes that I identified.) Examples of substantive errors include: (1) the derivation of the conditional distribution of one random variable given another when the underlying joint distribution is multinomial, (2) some CDF curves exceed unity, and (3) a statement made that no uniformly most powerful test exists for the one-sided case concerning a normal mean with known variance. In conclusion, Mathematical Statistics With Applications, after the many errors are corrected in a new printing, would be a reasonable textbook for the introductory mathematical statistics sequence at the graduate level. The many applications will be an aid to learning and any theoretical deficiencies can be supplemented in the classroom.